List of Notes - Category: Trigonometry

Notion of Complex Numbers

Process of extending notion of number from natural to real was connected both with needs of practice and requirements of mathematics.

At the beginning, people used numbers to count objects. Then when they began to divide things it appeared that they need more than natural numbers - positive rational numbers. Then, need in subtracting led people to 0 and negative numbers. Finally, need of taking roots of positive number led us to irrational numbers.

Algebraic Form of Complex Numbers

Complex number z=(a;b) can be represented in the form z=a+bi. This is called algebraic form of complex number.

In the representation z=a+bi, `i^2=-1` and `i` is called imaginary unit. a is real part (denoted by Re z) and b is imaginary part (denoted by Im z) of complex number.

Arithmetic operations over Complex Numbers

In general, arithmetic of complex numbers is same as arithmetic of real numbers, we just handle real and imaginary parts separately and remember that `i^2=-1` .

Sum of complex numbers `z_1=a+bi` and `z_2=c+di` is complex number `z_1+z_2=(a+bi)+(c+di)=(a+c)+(b+d)i`.

Geometric Representation of Complex Numbers. Trigonometric Form of Complex Numbers

Complex number z=a+bi on coordinate pllane xOy is represented by point M with coordinates (a,b). x-axis is called real axis and y-axis is called imaginary axis.

Real numbers are represented by points of real axis and purely imaginary numbers are represented by points of imaginary axis.

Operations over Complex Numbers in Trigonometric Form. De Moivre's Formula

Suppose that we are given two complex numbers in trigonometric form: `z_1=r_1(cos(phi_1)+isin(phi_1))` and `z_2=r_2(cos(phi_2)+isin(phi_2))`.

Then their product is `color(red)(z_1z_2=r_1r_2(cos(phi_1+phi_2)+isin(phi_1+phi_2)))` and their quotient is `color (blue)(z_1/z_2=r_1/r_2(cos(phi_1-phi_2)+isin(phi_1-phi_2)))`.

Converting Proper Fraction into Infinite Periodic Decimal

Suppose we have decimal fraction 2.73. If we add any number of zeros to the right, then value of fraction will not change: 2.73=2.730=2.7300=2.7300...0. This means that we can write the fraction 2.73 as decimal fraction with infinite set zeros i.e. 2.73=2.73000... . Here we have infinitely many decimal digits after dot. Such decimal fraction is called infinite decimal fraction.

Converting Infinite Periodic Decimal into Proper Fraction

If we need to multiply infinite decimal fraction by 10, 100, 1000 and so on we will shift a dot at one, two or three and so on digits to the right such as it do in the finite decimal fraction.

For example, `0.1(23)*100=0.1232323...*100=12.323232...=12.(32)` .

Number Plane.Cartesian Coordinate System in the Plane and Space

A pair of numbers is usually described as two numbers, which are consider in a specific order (an ordered pair).The set of all pairs of real numbers is called number plane.

As for the set of all real numbers (or number scale) is a geometric model such as the direct-axis and the set of all pairs of real numbers (number plane) is a geometric model such as the coordinate plane.

Coordinate Line

Let′s draw the line `l` and will note on it the point `O`, which we take as the origin. Thereafter we must to choose the direction and the unit segment.

In this case we say, that we have coordinate line. Every naturall number or fraction corresponds to one point of line `l`. Suppose, for example, it is 3. Let′s measure from the point `O` the unit segment three times in a given direction and we will get the point `A` - and it corresponds to the number 3. Analogically, if we have the number 4.2 we will measure it from point `O` the unit segment four times in a given direction and then we measure 0.2 more of the parts of this segment and we will obtain the point `B` - it corresponds to 4.2.

Polar Coordinate System

The position of a point in the plane can be set not only by its Cartesian coordinates `x, y` , but also in other ways. Let′s connect, for example, the point `M` with the origin `O` and consider next two numbers: the length of the segment `OM=r` and angle φ of tilt of this segment to the positive direction of the axis `Ox` (this angle will be positive if the rotation from the axis `Ox` to its shift with the direction `OM` is counterclockwise, and negative otherwise). The segment `r=OM` is called the polar radius of the point `M`, the angle `phi` is its polar angle, the pair of numbers `(r,phi)` are polar coordinates, the point `O` is pole, axis `Ox` is polar axis. Such coordinate system is called polar.

Properties of the Numerical Inequalities

For any real numbers `a, b, c, d` is used the following properties:

  1. If `a>b`, then `b<a`.
  2. If ` ` ` ``a>b` and `b>c`, then `a>c` (the transitive property).
  3. If `a>b` then `a+c>b+c`.
  4. If `a>b` and `c` - positive number (`c>0`), then `ac>bc` (if the both parts of correct inequality to multiply by the same positive number, then we obtain correct inequality)
  5. If `a>b` and c - negative number (`c<0`), then `ac<bc` (if the both parts of correct inequality to multiply by the same negative number and change the sign of the initial inequality in the opposite, then we obtain correct inequality).
  6. If `a>b` and `c>d`, then `a+c>b+d` ( if two correct inequalities add termwise, we will obtain correct inequality).
  7. If `a, b, c, d ` - positive numbers and `a>b` and `c>d`, then `ac>bd` ( if the correct inequality of same sign to multiply termwise, the left and right parts of which are positive numbers, we will obtain correct inequality).
  8. If `a>b` and `c<d`, then `a-c>b-d`.
  9. If `a>b>0`, then `1/a<1/b` .
  10. If `a>b>0`, then `a^n>b^n` for any naturall `n`.

Numerical Intervals

Let′s take two numbers `a` and `b` such, that `a<b` and point on the coordinate line corresponding points.

The set of all numbers `x`, that satisfies the inequalities `a<x<b` , designates (`a,b`) and called the interval.

Absolute Value of Real Numbers

Absolute value of real numbers `a` is called the number itself, if `a>=0` , and opposite number `-a`, if `a<0`. The absolute value of number is denoted by `|a|`.

Definition. Absolute value of a is `|a|={(a if a>=0),(-a if a<0):}` .

Formula for Distance Between Two Points

First, consider one-dimensional case. If `a` and `b` - are two points of coordinate line, then the distance between them `rho (a;b)` is expressed by the formula `rho (a;b)=|a-b|` . It is clear, that `rho(a;b)=rho(b;a)` . So, `rho(-2;5)=|-2-5|=|-7|=-(-7)=7`.

Integer Part of Numbers. Fractional Part of Number

Suppose, `x` is the real number.

Its integral part is the greatest integral number, that isn′t exceed `x`.

The integral part of number `x` denotes as `[x]`.

The fractional number of number `x` is difference between the number and its integral part, i.e. `x-[x]` .

The Power with Natural Exponent

Suppose, we have `a` is the real numbers, and `n` is natural numbers equals more than one; `n-th` power of the number `a` are product of `n` multipliers, each of them equal `a`, i.e.

The number `a` is the base of exponent, `n` is the power of exponent.

The Power with Zero Exponent. The Power with Negative Exponent

If `a!=0` , then `a^0=1` .

For example, `(2.7)^0=1; (-5)^0=1`.

The zero power of number ` `0 hasn′t meaning.

If `a!=0` and `n` - natural number, then `a^-n=1/a^n` .

For example, `5^-3=1/5^3=1/125; (-2)^-2=1/(-2)^2=1/4` .

The Root of Odd Degree n From Negative number a

Suppose `a<0` and `n` is natural number and is greatest than 1.

If `n` is even number, then equality `x^n=a` doesn′t execute for any real number `x`.

This means, that in the range of real numbers is impossible to determine the root of an even degree from a negative number.

The Properties of Powers with the Rational Exponents

For any number `a` is determined the operation of natural exponentiation; for any number `a!=0` is determined the operation of raising to the zero and integer negative power; for any number `a>=0` is determined the operation of raising to the positive fractional power; for any number `a>0` is determined the operation of raising to the negative fractional power.

Permutations

Permutations of k elements from n different elements are all possible arrangements, that contain k elements, taken from n given elements. Arrangements differ from each other if they contain different elements, or order of elements is different. For example, abc, bac, acd are different arrangements.

Arrangements

Arrangements of n different elements are all possible orderings of these n elements. Arrangements differ only by order of elements.

Arrangement is particular case of permutation when k=n.

There are two forms of arrangements:

Combinations and their Properties. Pascal's Triangle

Combinations of k elements from n different elements are all possible arrangements, that contain k elements, taken from n given elements. Arrangements differ from each other if they contain different elements. For example, abc, acd are different arrangements, but `abc,bca` are same combinations.

Binom of Newton

Let's write expressions for `(a+b)^n` when `n=1,2,3,4`:

`(a+b)^1=\ \ \ \ \ \ \ \ \ \ \ \ \ a+b`;

`(a+b)^2=\ \ \ \ \ \ \ \ a^2+2ab+b^2`;

`(a+b)^3=\ \ \ \ a^3+3a^2b+3ab^2+b^3`;

`(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4`.

Properties of Newton's Binom Formula

Properties for `(a+b)^n` are following:

  1. Number of all summands in expansion equals `n+1`.
  2. General member of expansion has form `T_(k+1)=C_n^ka^(n-k)b^k`, where k=0,1,2,...,n.
  3. Coefficients that are equidistant from ends of expansion are equal.
  4. Sum of all binomial coefficient is `2^n`.
  5. Sum of binomial coefficients on even positions equals sum of coefficients on odd positions.

Example 1. Find the greatest element in the expansion `(a+b)^n`, if sum of all binomial coefficients is 4096.

Basic Concepts Connected with Solving Inequalities

Suppose we are given inequality `f(x)>g(x)`. We call it inequality with one variable. Every value of variable that converts given inequality into correct numerical equality is called a solution of inequality. To solve inequality with one variable means to find all solutions of inequality or prove that there are no solutions.

Graphical Method for Solving Inequality with One Variable

To graphically solve inequality `f(x)>g(x)` we need to draw graphs of functions `y=f(x)` and `y=g(x)`, and choose those intervals on x-axis where graph of y=f(x) is higher then y=g(x).

Example. Solve graphically `log_2(x)>2/x`.

Linear Inequalities with One Variable

Here we talk about inequalities of the form `ax>b` or `ax<b,ax>=b,ax<=b`. If a>0 then inequality `ax>b` is equivalent to the inequality `x>b/a`, i.e. set of solutions of this inequality is interval `(b/a,+oo)`. If a<0 then inequality `ax>b` is equivalent to the inequality `x<b/a`, therefore, set of solutions of this inequality is interval `(-oo,b/a)`.

Systems of Inequalities with one Variable

We say that several inequalities form a system, if we need to find common solutions of given inequalities.

Value of variable for which every inequality in the system becomes correct is called a solution of the system of inequalities.

Set of Inequalities with One Variable

With say that several inequalities form a set, if we need to find solutions that satisfy at least one inequality.

Value of variable for which at least one inequality in the set becomes correct is called a solution of the set of inequalities.

Fractional-Linear Inequalities

Here, we will consider inequalities of the form `(ax+b)/(cx+d)>0` or `(ax+b)/(cx+d)<0`, where `a!=0,c!=0`.

Example. Solve inequality `(3x+7)/(2x-7)>5`.

In cases where right-hand side doesn't equal 0, we need to transform inequality into equivalent one, where right-hand side equals 0:

Quadratic Inequalities

Here we consider inequalities of the form `ax^2+bx+c>0` or `ax^2+bx+c<0`, where `a!=0`.

Fact. If discriminant `D=b^2-4ac` of quadratic function `ax^2+bx+c` is negative and `a>0`, then for all x `ax^2+bx+c>0`.

Solving Quadratic Inequalities Graphically

Graph of quadratic function `y=ax^2+bx+c` is a parabola, that is open upward if a>0, and open downward if a<0.

There are 3 possible cases:

  1. parabola intersects x-axis (i.e. equation `ax^2+bx+c=0` has two different roots);
  2. parabola has vertex on x-axis (i.e. equation `ax^2+bx+c=0` has one root);
  3. parabola doesn't intersect x-axis (i.e. equation `ax^2+bx+c=0` doesn't have roots).

Therefore, there are 6 possible positions of parabola, shown on figure.

Inequalities Involving Absolute Values

To solve solve inequalities that involve absolute values, we use following definition of absolute values:

`|f(x)|={(f(x) if f(x)>=0),(-f(x) if f(x)<0):}`.

Besides, we can use method of squaring of both sides of inequality, that is based on the following fact:

Solving Rational Inequalities Using Method of Intervals

Solving inequalities of the form `(p(x))/(q(x))>0` (instead of ">" there can be another symbol of inequality), where p(x) and q(x) are polynomials, is based on the following reasoning.

Consider function `h(x)=((x-a)(x-b))/((x-c)(x-d))`, where `a<b<c<d`. If `x>d`, then each of factors `x-a,x-b,x-c,x-d` is positive, therefore, on interval `(d,+oo)` we have that `h(x)>0`. If c<x<d, then `x-d<0`, and other factors are still positive. Therefore, on interval (c,d) we have that h(x)<0. Similarly on interval (b,c) we have that h(x)>0 etc. (whole process is shown on figure a)).

Exponential Inequalities

When we solve inequalities of the form `a^(f(x))>a^(g(x))` we need to remember, that exponential function `y=a^x` is increasing when a>1 and decreasing when 0<a<1.

Therefore, when a>1, `a^(f(x))>a^(g(x))` is equivalent to the inequality `f(x)>g(x)`. If 0<a<1, then `a^(f(x))>a^(g(x))` is euqivalent to `f(x)<g(x)`.

Logarithmic Inequalities

When we solve inequalities of the form `log_a(f(x))>log_a(g(x))` we need to remember, that logarithmic function `y=log_a(x)` is increasing when a>1 and decreasing when 0<a<1. Also, we need to remember, that logarithmic function is defined

Irrational Inequalities

When we solve irrational inequalities, we use follwoing fact:

Fact. If both sides of inequality on some interval X take only non-negative values then squaring both sides (or taking any even power) will give equivalent inequality (on interval X). Taking odd power of both sides will always give equivalent inequality.

Trigonometric Inequalities

We will consider here the simplest trigonometric inequality of the form f(x)<a (or f(x)>a), where f(x) is some trigonometric function.

Example. Solve `cos(x)<0.5`.

Let's draw graph of the function `y=cos(x)` and draw line `y=0.5`. We are interested in those intervals where cosine is below line y=0.5. One of such intervals is `(pi/3,(5pi)/3)`. Using periodicity of cosine we obtain solution: `pi/3+2pik<x<(5pi)/3+2pik,k in Z`.

Inequalities and System of Inequalities with Two Variables

Consider inequality `f(x,y)>g(x,y)`, which we will call inequality with two variables. Solution of such inequaltity is pair of value of arguments, that convert inequality into correct numerical identity.

Method of Estimating of the Sign of Difference

The essence of this method is in the following: to prove that `f(x,y,z)>g(x,y,z)` we need to prove that `f(x,y,z)-g(x,y,z)>0`.

Example. Prove that if `x>=0,y>=0` then `(x+y)/2>=sqrt(xy)` (arithmetic mean of two non-negative numbers is not less than their geometric mean; this inequality is called Cauchy's Inequality).

Synthetic Method of Proving Inequalities

The essence of this method is in following: with the help of the sequence of transformations we infer inequality from well-known inequalities.

Well-known inequalities are:

  1. `(x+y)/2>=sqrt(xy)`, where `x>=0,y>=0` (Cauchy's inequality);
  2. `x+1/x>=2`, where x>0;
  3. `|x+a|<=|x|+|a|` (triangle inequality);
  4. `-1<=sin(alpha)<=1`;
  5. `-1<=cos(alpha)<=1`.

Example. Show that `(a+b+c+d)/4>=root(4)(abcd)`.

Proving of Inequalities by Contradiction

The essence of this method is in the following: suppose we need to prove inequality `f(x,y,z)>g(x,y,z)`. We assume contrary proposition, i.e. that for some x,y and z `f(x,y,z)<=g(x,y,z)`.

Using properties of inequalities we perform transformations of inequality. If as result, we obtain incorrect inequality than this means that our assumption that `f(x,y,z)<=g(x,y,z)` was wrong, therefore, `f(x,y,z)>g(x,y,z)`.

Using Inequalities to Solve Equations

Suppose we need to solve equation `f(x)=g(x)` and suppose there exists such number A that is the maximum value of function `y=f(x)` and minimum value of function `y=g(x)`.

Then roots of the equation `f(x)=g(x)` are common roots of the equations `f(x)=A,g(x)=A` and only they.

Linear Equations

Equation of the form `ax=b`, where a and b are real numbers, is called linear equation with one variable x; a is called coefficient near variable, b is called free member.

For linear equation `ax=b` there are three possible cases:

Quadratic Equations

Equation of form `ax^2+bx+c=0` where a,b,c are real numbers and `a!=0` (if a=0 then equation becomes linear: `bx+c=0`), is called quadratic. If a=1, then quadratic equation is called reduced, if `a!=1` - equation is called unreduced. a is called first coefficient, b is called second coefficient, c is free member.

Systems and Sets of Equations

Suppose we are given two equations `f_1(x)=g_1(x)` and `f_2(x)=g_2(x)`. If we need to find values of variable that satisfy both given equations, it is said that system of equations is given: `{(f_1(x)=g_1(x)),(f_2(x)=g_2(x)):}`.

Equations Involving Absolute Values

Example 1. Solve `|2x-7|=3`.

If `|a|=3` then either a=3 or a=-3. This means that given equation is equivalent to the set of equations: `2x-7=3,2x-7=-3`.

From first equation we find that x=5, from second we find that x=2. Thus, there are 2 solutions: 2 and 5.

Notion of the Consequence of Equation. Extraneous Roots.

Suppose we are given two equations `f_1(x)=g_1(x)` and `f_2(x)=g_2(x)`.

If every root of first equation is also root of the second equation, then second equation is called consequence of the first equation.

Equations with Variable in Denominator

Consider equation of the form `(p(x))/(q(x))=0`.

Solving of this equation is based on the following fact: fraction equals zero if and only if numerator equals zero and denominator doesn't equal zero (we can't divide by 0).

Rational Equations

Equation `f(x)=g(x)` is called rational, if both f(x) and g(x) are rational expressions. If f(x) and g(x) are integer expressions, then equation is called integer; if at least one of expressions f(x), g(x) is fractional, then rational equation `f(x)=g(x)` is called fractional.

Solving of Equation p(x)=0 by Factoring Its Left Side

The essence of method of factoring is following. Suppose we need to solve equation p(x)=0, where p(x) is polynomial of n-th degree. Suppose we factored it as `p(x)=p_1(x)*p_2(x)*p_3(x)`, where `p_1(x),p_2(x),p_3(x)` are polynomials of lesser degree than n. Then, instead of solving equation p(x)=0, we need to solve set of equations `p_1(x)=0,p_2(x)=0,p_3(x)=0`. All found roots of these equations, and only they, will be roots of the equation p(x)=0.

Solving of Equations with Method of Introducing New Variable

It is easier to explain essence of the method of introducing new variable on example.

Example 1. Solve equation `(x^2-3x)^2+3(x^2-3x)-28=0`.

Let `x^2-3x=y`, then initial equation can be rewritten as `y^2+3y-28=0`. This equation has two roots: `y_1=-7,y_2=4`.

Biquadratic Equation

Equation of the form `ax^4+bx^2+c=0`, where `a!=0` is called biquadratic. Biquadratic equations is solved with method of introducing new variable: setting `x^2=y`, we obtain quadratic equation `ay^2+by+c=0`.

Equations of Higher Degrees

Consider equation of the form P(x)=0, where P(x) is polynomial of degree higher than 2.

To solve such equations method of factoring and method of entering new variable are used.

Factoring of polynomial of n-th degree is based on the following fact.

Solving of Word Problems with the Help of Equations

With the help of equations we can solve many various problems from physics, economics and other sciences.

Let's describe the general scheme for solving such problems:

  1. Introduce new variables, i.e. denote unknown quantities which we need to find (or they are needed to find required quantitie) by letters x,y,z.
  2. Using introduced variables and conditions in the problem, set up system of equations (or one equation).
  3. Solve system of equations (or one equation) and from obtained solution choose only those solutions that make sense according to problem.
  4. If required quantities are not denoted by x,y,z, then using obtained solutions, find answer to the problem.

Example 1. Two workers, working together, can do some work in 6 hours. First worker, working alone, can do all work 5 hours than second worker, if second worker works alone. How many hours does each of them need to complete the work, working separately?

Irrational Equations

Irrational equation is equation, that contains variable under radical or variable is a base of power with fractional exponent. For example, `sqrt(x-2)=2x-1` and `x^(1/4)-5=0` are irrational equations.

Exponential Equations

Equations of the form `a^(f(x))=a^(g(x))`, where `a>0,a!=1` are called exponential equations.

To solve such equation we use following property: exponential equation `a^(f(x))=a^(g(x))` is equivalent to the equation `f(x)=g(x)`.

Logarithmic Equations

Equations of the form `log_a(f(x))=log_a(g(x))`, where `a>0,a!=1` are called logarithmic equations.

To solve such equation we use following property: logarithmic equation `log_a(f(x))=log_a(g(x))` has equation-consequence `f(x)=g(x)`.

Exponential Logarithmic Equations

Example. Solve equation `x^(1-text(lg)(x))=0.01`.

Domain of the equation is x>0.

In this domain both left and right parts take only positive values, therefore we can take logarithm of both sides: `text(lg)(x^(1-text(lg)(x)))=text(lg)(0.01)` or `(1-text(lg)(x))text(lg)(x)=-2`.

The Simplest Trigonometric Equations

Equation `sin(x)=a` where `|a|<=1` has infinitely many roots. For example roots of the equation `sin(x)=1/2` are following: `x_1=(pi)/6,x_2=(5pi)/6,x_3=pi/6+2pi,x_4=pi/6-2pi` etc. The common formula, that allows to find all roots of the equation `sin(x)=a` is following:

Solving of Trigonometric Equations by Factoring

If trigonometric equation has form f(x)=0, and left side f(x) can be factored, then we need to equal to zero all factors. As result, we will obtain set of equations, and roots of each of them will be roots of the initial equation if they belong to the domain of each factor in the left part of equation.

Solving of Trigonometric Equatipns with a Method of Introducing New Variable

The essence of this method is in following: we try to transform given trigonometric equation into algebraic equation by introducing new variable.

Example 1. Solve equation `2cos^2(x)+14cos(x)=3sin^2(x)`.

Homogeneous Trigonometric Equations

In many cases trigonometric equation of the form `R(sin(x),cos(x)=0)` can be transformed into algebraic with respect to `tan(x)`. Main property of such equations is that they don't change if we replace `sin(x)` with `-sin(x)` and `cos(x)` with `-cos(x)`.

Universal Substitution for Rational Trigonometric Equations

It is know that sin(x) and cos(x) can be rationally expressed through `tan(x/2)`, namely:

`sin(x)=(2tan(x/2))/(1+tan^2(x/2)),\ cos(x)=(1-tan^2(x/2))/(1+tan^2(x/2))`, where `x!=pi+2pin, n in Z` (i.e. where `tan(x/2)` is not defined).

Method For Solving Trigonometric Equations by Introducing Auxiliary Argument

Sometimes it is useful to replace expression `acos(x)+bsin(x)` with expression `A cos(x+phi)`, where `A=sqrt(a^2+b^2)`, `sin(phi)=a/(sqrt(a^2+b^2))`, `cos(phi)=b/(sqrt(a^2+b^2))`. In this case `phi` is called auxiliary argument.

Graphical Solving of Equations

On practice it is often useful to use graphical method to solve equations. The essence of this method is in following: to solve equation f(x)=0 we draw graph of the function y=f(x) and find x-intercepts; these points will be roots of the equation.

Parametric Equations

Suppose we are given equality with variable x and a: `f(x,a)=0`. If we need for real value of a solve this equation with respect to x, then equation `f(x,a)=0` is called equation with variable x and parameter a. To solve equation with parameter a means for every a to find x, that satisfy given equation.

Solving of Equation with Two Variables

Consider equation with two variables `f(x,y)=0`.

Pair of values that makes from equation correct equality is called solution of the equation. If we are given equation with two variables x and y, then it is a rule to write its solution as (x;y) (x is on first place and y is on second).

Graph of the Function with Two Variables

Suppose we are given equation with two variables `f(x,y)=0`. If we draw all its solutions on coordinate plane, then we obtain some set of points on plane. This set is called graph of the equation `f(x,y)=0`.

Linear Equation with Two Variables and Its Graph

Equation of the form `ax+by=c`, where x,y are variables and a,b,c are numbers, is called linear; numbers a adn b are called coefficients near variables, number c is called free member.

Graph of any linear equation `ax+by=c`, where at least one coeffcient doesn't equal zero, is line; if a=0, then line is parallel to the x-axis, if b=0, then it is parallel to th y-axis (see figure to the right).

Systems of Two Equations with Two Variables. Equivalent Systems

Suppose we are given two equations with two variables `f(x,y)=0` and `g(x,y)=0`. If we need to find all common solutions of two equations with two variables, then it is said that system of equations should be solved. Every pair of values that satisfy all equations in the system, is called a solution of the system of equations.

Solving of System of Two Equation with Two Variables. Substitution Method

Substitution method consists of following steps:

  1. Transform one of the equations into the form where variable y is expressed in terms of x (or x in terms of y).
  2. In another equation replace y (or x) with obtained expression. As result you will obtain equation with one variable
  3. Find roots of this equation.
  4. Using expression for y in terms of x (or x in terms of y) find corresponding values of y (or x).

Example. Solve system of equations `{(x-3y=10),(x^2-24y=100):}`.

Solving of System of Two Equation with Two Variables. Addition Method

This method is based on facts about equivalent systems.

Example. Solve system of equations `{(2x+3y=7),(3x-y=16):}`.

Multiplying both sides of second equation by 3, we obtain equivalent system: `{(2x+3y=7),(9x-3y=48):}`.

Solving of System of Two Equation with Two Variables. Method of Introducing New Variables

There are two ways to use method if introducing new variables:

  1. introduce one new variable only for one equation of the system;
  2. introduce two new variables for both equations of the system.

Example 1. Solve system `{(x/y+y/x=13/6),(x+y=5):}`.

System of Two Linear Equations with Two Variables. Second-Order Determinants

In theory of systems of linear equations it is convenient to use notion of determinant.

Second-order determinant is a number, that is defined by equality `|[a_(11),a_(12)],[a_(21),a_(22)]|=a_(11)a_(22)-a_(21)a_(12)`.

Symmetric Systems

Polynomial P(x,y) is called symmetric if interchanging of x and y doesn't change polynomial, i.e. P(x,y)=P(y,x).

For example, polynomial `P_1(x,y)=x^3+3xy+y^3` is symmetric because `P_1(y,x)=y^3+3yx+x^3=x^3+3xy+y^3=P_1(x,y)`, and `P_2(x,y)=x^2+y` is not because `P_2(y,x)=y^2+x !=P_2(x,y)`.

Graphical Solving of the System of Two Equations with Two Variables

To solve graphically system of two equations with two variables, we need to draw graphs of the equations and find coordinates of points of intersections of these graphs.

Example. Solve the following system graphically: `{(x^2+y^2=25),(xy=12):}`.

Systems of Three Equations with Three Variables

Consider system of three equations with three variables `{(f(x,y,z)=0),(g(x,y,z)=0),(h(x,y,z)=0):}`.

Solution of such system is any triple of numbers that satisfies all equations in the system.

Example 1. Solve system equations `{(x+y+z=2),(2x+3y+z=1),(x^2+(y+2)^2+(z-1)^2=9):}`.

Systems of Three Linear Equations with Three Varaibles. Third-Order Determinants

Third-order determinant is a number that is defined by equality `|[a,b,c],[d,e,f],[g,h,i]|=aei+bfg+cdh-ceg-afh-bdi`.

This formula is a bit hard, but luckily you don't need to remember it. It is sufficiently to understand how this formula is derived.

Systems of Exponential and Logarithmic Equations

When we solve systems of exponential and logarithmic equations, we use standard techniques for solving exponential and logarithmic equations, and standard techniques for solving system of equations.

Example. Solve the following system of equations: `{(log_2(x)+log_4(y)=4),(3^(x^2)=9*3^(15y+2)):}`.

Systems of Trigonometric Equations

Whe we solve systems of trigonometric equations, we use trigonometric formulas and standard techniques for solving systems of equations.

Example. Solve the following system of equations: `{(sin(x)+cos(y)=1.5),(sin^2(x)+cos^2(y)=1.25):}`.

Approximate Values of the Number. Absolute and Relative Errors

When we round off decimal to some digit, all digits after this digit will be replaced by zero, and if they are after the dot, then they will be deleted. If the first next digit after this digit is more than or equal 5, then the last remaining digit will be not changed.

Decimal Approximations of the Real Number by Excess and Defect

Let′s take the irrational number `sqrt(2)` and we have:

`1^2<2<2^2` ; `1<sqrt(2)<2` ;

`1.4^2<2<1.5^2` ; `1.4<sqrt(2)<1.5` ;

`1.41^2<2<1.42^2` ; `1.41<sqrt(2)<1.42` ;

The Degree with the Irrational Exponent

Let′s suppose `a` is irrational number. What is the sense of record `a^a` , where `a` - positive number? Let′s consider three events: `a=1` , `a>1` , `0<a<1` .

  1. If `a=0` , then we suppose `1^a=1` .
  2. Suppose `a>1`. Let′s take any rational number `r_1<a` and any rational number `r_2>a`. Thereat `r_1<r_2` and `a^(r_1)<a^(r_2)` . In this case `a^a` means the number, that is between `a^(r_1)` and `a^(r_2)` for any rational numbers `r_1` and `r_2` such, that `r_1<a` and `r_2>a`. It is proved, that there is such number and singular for any `a>1` and for any irrational number `a`.
  3. Suppose `0<a<1`. Let′s take any rational number `r_1<a` and any rational number `r_2>a`. Thereat `r_1<r_2` and `a^(r_1)>a^(r_2)` . In this case `a^a` means such number, that is between `a^(r_2)` and `a^(r_1)` for any rational numbers `r_1` and `r_2`, satisfying the inequality `r_1<a<r_2`. It is proved, that there is such number and singular for any number `a` with the interval `(0,1)` and any irrational number `a`.

Definition of the Function

Numerical function with domain D is a correspondence, when for every number x from set D we find by some rule number y, that depends on x.

Variable x is called independent variable (or argument). Number y, that corresponds to number x, is called value of a function f at point x and is denoted by f(x). We denote by letter f given function, i.e. functional dependence between variables x and using record y=f(x).

Analytical Representation of the Function

To define function, we need to specify way, that allows us for every value of argument find corresponding value of the function. The most common way to set function is to represent it with the help of formula y=f(x), where f(x) is some expression with variable x. In this case it is said that function is represented by formula or function is represented analytically.

Tabular Representation of the Function

On practice tabular representation of the function is often used. This is because in many real-world situations we can't represent functional dependence analytically.

In this case table is formed where we specify values of the function for any x in table.

Graphical Representation of the Function

Let's take rectangular Descartes coordinate system; let's draw on coordinate plane all points with x-coordinate x=a, we will obtain line, that is parallel to the x-axis; we say that x=a is equation of this line, in particular, x=0 is equation of y-axis. Similarly, let's draw on coordinate plane all points with y-coordinate y=b, we will obtain line, that is parallel to the x-axis; it is said that y=b is equation of this line, in particular y=0 is equation of x-axis.

Graph of the Function Represented Analytically

Suppose we are given function that is represented analytically by the formula y=f(x).

Then its graph is a set of all points (x;y), where y=f(x), and x takes all values from the domain of f.

Example 1. Draw graph of y=x.

Odd and Even Functions

Function y=f(x) is called even if for any x from domain of f we have that `f(-x)=f(x)`.

For example, `y=x^2` is even function because `f(-x)=(-x)^2=x^2=f(x)`. Another examples of even functions are `y=x^4` and `y=x^6`.

Graphs of Even and Odd Functions

Graphs of even and odd functions have following properties:

  1. If function is even then its graph is symmetric about y-axis.
  2. If function is odd then its graph is symmetric about origin.

Example 1. Draw graph of the function `y=|x|`.

Periodic Functions

Function y=f(x) is called periodic, if exists such number `T!=0`, that for any x from domain of the function we have the following: `f(x+T)=f(x)=f(x-T)`.

Number T is called period of the function y=f(x).

Constant Function

Function is constant if it has form `y=b`, where b is some number.

Graph of constant function y=b is line, parallel to the x-axis, that passes through point (0;b) on y-axis.

Direct Proportionality

It is said that y is directly proportional to x if their ratio is constant, i.e. `y/x=k` or `y=kx`.

Direct proportionality is a function of the form `y=kx`, where `k!=0`. Number k is called coefficient of proportionality.

Relative Position of Graphs of the Linear Functions

Suppose we are given two linear functions `y=k_1x+b_1` and `y=k_2x+b_2`.

Graphs of these functions are lines.

These lines intersect if `k_1!=k_2`. In particular, if `k_1k_2=-1` then lines are perpendicular.

Inverse Proportionality

It is said that y is inversely proportional to x if their product is constant, i.e. `xy=k` or `y=k/x` .

Inverse proportionality is a function of the form `y=k/x`, where `k!=0`. Number k is called coefficient of inverse proportionality.

Function `y=x^2`

Properties of the function `y=x^2`:

  1. Domain is all number line, i.e. interval `(-oo,+oo)`.
  2. Function is even: `f(-x)=(-x)^2=x^2=f(x)`.
  3. On interval `[0,+oo)` function is increasing.
  4. On interval `[-oo,0]` function is decreasing.

Graph of this function is parabola.

Function `y=x^3`

Properties of the function `y=x^3`:

  1. Domain is all number line, i.e. interval `(-oo,+oo)`.
  2. Function is odd: `f(-x)=(-x)^3=-x^3=-f(x)`.
  3. Function is increasing on all number line.

Graph of this function is called cubic parabola.

The Types of Algebraic Expression

From the numbers and variables with the helping of sign of addition, subtraction, multiplication, division, raising to rational power and root extraction and with the helping of bracket we can compose the algebraic expression.

Allowable Value of Variables. Domain of Algebraic Expression

The values of variables, for which the algebraic expression makes sense, are called allowable values of variables. The set of all allowable values of variables is called domain of algebraic expression.

The Concept of Identity Transformation Expression. Identity

Let′s consider two expressions: `f(x)=x^2-2x` and `g(x)=4x-5`. For `x=2` we have `f(2)=2^2-2*2=0`; `g(x)=4*2-5=3`. The numbers 0 and 3 are called corresponding values of expressions `x^2-2x` and `4x-5` for `x=2`.

Monomials and Operations on them

Monomial is caled such expression, that contains the numbers, natural powers of variables and their products and doesn′t contain any other operations on numbers and variables.

For example, `3a*(2.5a^3)`, `(5ab^2)*(0.4c^3d)`, `x^2y*(-2z)*0.85` - monomials, whereas the expressions `a+b, (ab)/c` aren′t monomials.

Polynomials. Transformation of Polynomials to the Standart Form

The sum of monomials are called polynomial.

If we all terms ofpolynomial write in the standart form and transform the similar terms, then we will obtain the polyinomial of standart form.

Any integral expression we can transform to the polynomial of standart form -it is purpose of transformation (simplification) of integral expressions.

Short Multiplication Formulas

In some cases the transformation of integral expression into the standart form of polynomial realize with the using of identities:

  1. `(a+b)(a-b)=a^2-b^2`
  2. `(a+b)^2=a^2+2ab+b^2`
  3. `(a-b)^2=a^2-2ab+b^2`
  4. `(a+b)(a^2-ab+b^2)=a^3+b^3`
  5. `(a-b)(a^2+ab+b^2)=a^3-b^3`
  6. `(a+b)^3=a^3+3a^2b+3ab^2+b^3`
  7. `(a-b)^3=a^3-3a^2b+3ab^2-b^3`

These identities are called short multiplication formulas; the formula 1- difference of squares; the formulas 2 and 3 - respectively the square of sum and the square of difference, the formulas 4 and 5 - the sum of cubes and difference of cubes, and formulas 6 and 7- the cube of sum and cube of difference.

Power Function with Natural Exponent

Function `y=x^n`, where n is natural number, is called power function with natural exponent.

When n=1, we obtain function y=x; when n=2, we obtain parabola `y=x^2`; when n=3, we obtain cubic parabola `y=x^3`.

Power Function with Integer Negative Exponent

Consider function `y=x^(-n)`, where n is natural number.

When n=1, we obtain that `y=x^(-1)` or `y=1/x`. This is hyperbola.

Let n is odd number greater than one: n=3, 5, 7, ... . In this case function `y=x^(-n)` has same properties as `y=1/x`. Graph of the function `y=x^(-n)` (n=3, 5, 7, ...) resembles graph of the function `y=1/x` (see left figure). If |x|<1 the the bigger n, the further graph from x-axis. If |x|>1 the the bigger n, the closer graph to x-axis.

Function `y=sqrt(x)`

Properties of the function `y=sqrt(x)`:

  1. Domain is interval `[0,+oo)`.This is because expression `sqrt(x)` makes sense only for `x>=0`.
  2. Function is neither even, nor odd.
  3. Function is increasing on `[0,+oo)`.

To draw graph of the function let's find some values of function: if x=0, `y=sqrt(0)=0`; if x=1, `y=sqrt(1)=1`; if x=2, `y=sqrt(4)=2`; if x=3, `y=sqrt(9)=3`.

Function `y=root(3)(x)`

Properties of the function `y=root(3)(x)`:

  1. Domain of the function is all number line, i.e. interval `(-oo,+oo)`.
  2. Function is odd: `f(-x)=root(3)(-x)=-root(3)(x)=-f(x)`.
  3. Function is increasing on all number line.

To draw graph of the function let's find some values of function for `x>=0`: if x=0, `y=root(3)(0)=0`; if x=1, `y=root(3)(1)=1`; if x=4, `y=root(3)(4)~~1.6`; if x=8, `y=root(3)(8)=2`.

Factoring Polynomials

Sometimes we can transform the polynomials into product of several multipliers - polynomials or monomials. Such identical transformation is called factoring polynomials.

In this case we say, that polynomial is divided into each of these factors.

Function `y=root(n)(x)`

When n is even, function `y=root(n)(x)` has same properties as `y=sqrt(x)`. Graph of then function `y=root(n)(x)` resembles graph of the function `y=sqrt(x)`. When |x|<1 the bigger n, the further graph from the x-axis. When |x|>1 the bigger n, the close graph to the x-axis (see left figure).

Power Function with Positive Fractional Exponent

Consider function `y=x^r`, where r is positive irreducible fraction.

Properties of this function are following:

  1. Domain is interval `[0,+oo)`.
  2. Function is neither even, nor odd.
  3. Function is increasing on interval `[0,+oo)`.

On the left figure is drawn graph of the function `y=x^(5/2)`. It is located between graphs of the function `y=x^2` and `y=x^3`, defined on interval `[0,+oo)`. Similar form has any graph of the function `y=x^r`, where `r>1`.

Power Function with Negative Fractional Exponent

Consider function `y=x^(-r)`, where r is positive irreducible fraction.

Properties of this function are following:

  1. Domain is interval `(0,+oo)`.
  2. Function is neither even, nor odd.
  3. Function is increasing on `(0,+oo)`.

On the figure is shown graph of the function `y=x^(-1/2)`. Similar for has graph of any function `y=x^r`, where r is negative fraction.

Function y=[x] (integer part of number)

Recall, that `[x]` is the biggest integer that is less or equal than x. For example, `[0.1]=0,[1.9]=1,[-1.4]=-2`.

If `0<=x<1`, then `y=[x]=0`; if `1<=x<2`, then `y=[x]=1`; if `2<=x<3`, then `y=[x]=2` etc.

Function y={x}

Recall that `{x}=x-[x]`., where [x] is the greatest integer that is less or equal than x.

Note, that for any x we have that `{x-1}={x}={x+1}`.

This means that function `y={x}` is periodic with period T=1.

Polynomial of One Variable

The polynomial `ax+b` , where `a,b` are numbers `(A!=0)` and `x` is variable, is called first degree polynomial; the polynomial `ax^2+bx+c` , where `a, b, c` are numbers `(a!=0)` and `x` is variable, is called second degree polynomial (or trinomial square); the polynomial `ax^3+bx^2+cx+d`, where `a, b, c, d` are numbers `(a!=0)` and `x` is variable, is called third degree polynomial.

Division of Polynomials. Horner's Scheme. Bezout's Theorem

We can add, subtract, multiply and raise to the natural power the polynomials. We can divide the polynomial by polynomial sometimes.

If there is such polynomial `S(x)` , that `P(x)=Q(x)S(x)` , then we say, that the polynomial `Q(x)` divides polynomial `P(x)` and `P(x)` is called dividend, `Q(x)` is divisor and `S(x)` is quotient.

Inverse Function. Graph of the Inverse Function

Let's compare two functions y=f(x) and y=g(x) (see figure). They are both defined on segment [a,b], and their range is segment [c,d]. First function has the following property: for any `y_0` from segment [c,d] there is ONLY ONE value `x_0` from segement [a,b] such that `f(x_0)=y_0`. Geometrically this means that any horizontal lines that intersects y-axis between points c and d, intersects graph of the function y=f(x) only once. Second function doesn't have this property: for example, for value `y_1` line `y=y_1` intersects graph of the function y=g(x) three times. Therefore, in first case for any fixed value `y_0` from segment [c,d] equation `f(x)=y_0` has only one root `x_0`; in second case for some values of y, for example for `y=y_1`, equation `g(x)=y_1` has more than one root.

Logarithmic Function

Exponential function `y=a^x` has all properties that guarantee existence of inverse function:

  1. Domain is all number line.
  2. Range is interval `(0,+oo)`.
  3. Function is increasing when a>0 and decreasing when 0<a<1.

These properties guarantee existence of function, that is inverse to exponential. This function is defined on `(0,+oo)` and its range is all number line.

Factoring Quadratic Polynomials into Linear Factors

If `x_1` and `x_2` are the roots of quadratic polynomial of `ax^2+bx=c` (i.e.the root of equation of `ax^2+bx+c=0`), then `color(blue)(ax^2+bx+c=a(x-x_1)(x-x_2))`.This is the formula of factoring quadratic polynomial into factors.

Factoring Binomials `x^n-a^n`

It is known, that

`color(red)(x^2-a^2=(x-a)(x+a))`,

`color(blue)(x^3-a^3=(x-a)(x^2+xa+a^2))`.

When we multiply the polynomials `x-a` by `x^3+x^2a+xa^2+a^3`, we will obtain `color(green)(x^4-a^4=(x-a)(x^3+x^2a+xa^2+a^3))`.

Number `e`. Function `y=e^x`. Function `y=ln(x)`

Among exponential function `y=a^x`, where a>1, special interest for math and its applications is function that has following property: tangent line to the graph of the function at point (0;1) forms with x-axis `45^0` angle (see left figure). Base a of such function `y=a^x` is denoted by letter e, i.e. `y=e^x`. It is calculated that `e=2.718281824590...`. e is irrational number and can be represented as following sum: `e=1+1/1+1/(1*2)+1/(1*2*3)+...+1/(1*2*3*...*n)+...`.

Raising Binomial to the Natural Power (Newton's Binom Formula)

Here we will talk about how to raise `a+b` to any natural power.

if `n=1`, then `(a+b)^1=a+b`.

If `n=2`, then `(a+b)^2=a^2+2ab+b^2`.

If `n=3`, then `(a+b)^3=a^3+3a^2b+3ab^2+b^3`.

Using the fact that `(a+b)^4=(a+b)^3xx(a+b)`, we can write the following formula `(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4`.

Rational Fraction and its Basic Property

We can write any fractional algebraic expression in the form of `P/Q` , where `P` and `Q` are rational expressions, besides `Q` obligatory contains the variables. Such fraction is called rational fraction.

Reducing of Rational Fractions

To reduce fraction means to divide the numerator and denominator by common multiplier.

Posibility of such reduction is conditional by basic property of fraction.

In order to reduce rational fraction, we should factoring the numerator and denominator.

Reducing Rational Fractions to the Common Denominator

The common denominator of few rational fractions is integral rational expression, that is divided by denominator of each fraction (see note factoring polynomials).

We usually take such common denominator, that any another common denominator is divided by chosen. So, the common denominator of fractions `x/(x+2)` and `(3x-1)/(x-2)` is polynomial `(x+2)(x-2)`. We have `x/(x+2)=(x(x-2))/((x+2)(x-2))` ; `(3x-1)/(x-2)=((3x-1)(x+2))/((x+2)(x-2))` .

Definition of Trigonometric Functions

Trigonometric functions are defined with the help of coordinates of rotating point. Consider on x-y plane a circle with radius 1 and centered at origin (it is also called unit circle). Let's denote by `P_0` a point on a circle with coordinate (1;0); this point is called starting point. Let's take arbitrary number t and rotate starting point on angle t about point O (origin); if t>0 then we rotate counter-clockwise, if t<0 then we rotate clockwise.

Signs of Trigonometric Functions

Since `sin(t)` and `cos(t)` are respectively y-coordinate and x-coordinate of point `P_t`, that lies on unit circle centered at origin, then sine is positive for points that lie on upper half of the circle and negative for points that lie on lower half of the circle. Cosine is positive for points that lie on right half of the circle and negative for points that lie on left half of the circle.

Even and Odd Trigonometric Functions

Points `P_t` and `P_(-t)` of unit circle has same x-coordinate and same y-coordinates with opposite signs (see figure).

This means that `cos(-t)=cos(t)`, `sin(-t)=-sin(t)`, i.e. function `y=cos(x)` is even and function `y=sin(x)` is odd. From definition it follows that functions `y=tan(x)` and `y=cot(x)` are odd.

Periodicity of Trigonometric Functions

Since `P_t` and `P_(t+360^0)` is same point on unit circle, then sines of corresponding angles are equal. Same can be said about cosines.

Therefore, `color(red)(sin(x+360^0)=sin(x))`, `color(blue)(cos(x+360^0)=cos(x))`.

Properties and Graph of the Function y=sin(x)

Properties are following:

  1. Domain is all number line.
  2. Range is segment `[-1,1]`.
  3. Function is periodic; main period is `2pi`.
  4. Function is odd.
  5. Function is increasing on intervals `[-pi/2+2pin,pi/2+2pin]` and decreasing on intervals `[pi/2+2pin,(3pi)/2+2pin],n in Z` (see figure).

Let's first draw graph on the interval `[0,pi]`.

Multiplication and Division of Rational Fractions

Product of two ( and in general any finite number) rational fraction equals the fraction with the numerator, that equals the production of denominators of multiplied fractions: `color(red)(P_1/Q_1*P_2/Q_2=(P_1*P_2)/(Q_1*Q_2))` .

Properties and Graph of the Function y=cos(x)

Properties are following:

  1. Domain is all number line.
  2. Range is segment `[-1,1]`.
  3. Function is periodic with main period `2pi`.
  4. Function is even.
  5. Function is decreasing on intervals `[2pin,pi+2pin]` and increasing on intervals `[-pi+2pin,2pin],n in Z`.

Let's first draw graph on the interval `[0,pi]`. Find some values of function:

Properties and Graph of the Function y=tan(x)

Properties are following:

  1. Domain is `x!=pi/2+pik,k in Z` (in other words function is not defined for those values of x where `cos(x)=0`).
  2. Range is all number line.
  3. Function is periodic with main period `pi`.
  4. Function is odd.
  5. Function is increasing on intervals `[-pi/2+pik,pi/2+pik],k in Z`.
  6. Lines `x=pi/2+pik,k in Z` are vertical asymptotes.

Let's first draw graph on the interval `[0,pi/2)`. Find some values of function:

Properties and Graph of the Function y=cot(x)

Properties are following:

  1. Domain is `x!=pik,k in Z` (in other words function is not defined for those values of x where `sin(x)=0`).
  2. Range is all number line.
  3. Function is periodic with main period `pi`.
  4. Function is odd.
  5. Function is increasing on intervals `[pik,pi+pik],k in Z`.
  6. Lines `x=pik,k in Z` are vertical asymptotes.

Let's first draw graph on the interval `(0,pi/2]`. Find some values of function:

Raising Rational Fraction to the Integer Power

In order to raise rational fraction `P/Q` to the natural `n`-power, we should raise to this power the numerator and denominator of fraction separately; the first expression is numerator and second expression is denominator of the result: `color(blue)((P/Q)^n=P^n/Q^n)` .

Function y=arcsin(x)

Function y=sin(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=sin(x)` more than once.

However, if we consider interval `[-pi/2,pi/2]` then on this interval function passes horizontal line test. On this interval function is increasing and takes all values between -1 and 1. Therefore, for function `y=sin(x),-pi/2<=x<=pi/2` there exists inverse function. This function is denoted by `y=arcsin(x)` (sometimes also denoted by `y=asin(x)` or `y=sin^(-1)(x)`) and is read as arcsine.

Function y=arccos(x)

Function y=cos(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=cos(x)` more than once.

However, if we consider interval `[0,pi]` then on this interval function passes horizontal line test. On this interval function is decreasing and takes all values between -1 and 1. Therefore, for function `y=cos(x),0<=x<=pi` there exists inverse function. This function is denoted by `y=arccos(x)` (sometimes also denoted by `y=acos(x)` or `y=cos^(-1)(x)`) and is read as arccosine.

Function y=arctan(x)

Function y=tan(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=tan(x)` more than once.

However, if we consider interval `(-pi/2,pi/2)` then on this interval function passes horizontal line test. On this interval function is increasing and takes all values. Therefore, for function `y=tan(x),-pi/2<x<pi/2` there exists inverse function. This function is denoted by `y=arctan(x)` (sometimes also denoted by `y=atan(x)` or `y=tan^(-1)(x)`) and is read as arctangent.

Function y=arccot(x)

Function y=cot(x) is not invertible (not one-to-one), because it fails horizontal line test. We can draw such horizontal line that it will intersect graph of `y=cot(x)` more than once.

However, if we consider interval `(0,pi)` then on this interval function passes horizontal line test. On this interval function is decreasing and takes all values. Therefore, for function `y=cot(x),0<x<pi` there exists inverse function. This function is denoted by `y=text(arccot)(x)` (sometimes also denoted by `y=acot(x)` or `y=cot^(-1)(x)`) and is read as arccotangent.

Drawing Graph of the Function y=mf(x)

Task 1. Draw graph of the function `y=mf(x)`, where `m>0,m!=1`, knowing graph of the function `y=f(x)`.

We obtain y-coordinates of points of the graph of the function `y=mf(x)` by multiplying corresponding y-coordinates of points of the graph of the function `y=f(x)` by number m. Such transformation of graph of the function `y=f(x)` is called stretching from x-axis with coeffcient m if m>1, and compressing to x-axis, if 0<m<1.

Transformation of Rational Expressions

Transformation of any rational expression reduces to addition, subtraction, multiplication and division, raising to natural power of rational fractions. We can transform any rational expression into fraction with the numerator and denominator, that are integer rational expressions; this is sense of identical transformation of rational expression.

Graphs of Functions `y=ax^2,y=ax^3`

Graph of the function `y=x^2` is parabola. To draw graph of the function `y=ax^2` we need to stretch (compress) parabola `y=ax^2` from x-axis with coeffcient `|a|`; if a<0, then we addtionally need to reflect graph of the function `y=|a|x^2` about x-axis.

Drawing Graph of the Function y=f(x-a)+b

To draw graph of the function `y=f(x)+b` we need to shift graph of the function `b` units up if b>0, or `b` units down if b<0 (see figure to the right).

To draw graph of the function `y=f(x-a)` we need to shift graph of the function `a` units right if a>0, or `a` units left if a<0 (see figure to the left).

Graph of the Quadratic Function

Quadratic function is function of the form `y=ax^2+bx+c` where `a,b,c` are arbitrary constants and `a!=0`.

To draw graph of this function we need to perform following transformations of function (also known as "completing the square"): `ax^2+bx+c=a(x^2+b/a x)+c=a((x^2+2* b/(2a) x+(b^2)/(4a^2))-(b^2)/(4a^2))+c=a((a+b/(2a))^2-(b^2)/(4a^2))+c=`

Ways to Draw Graph of the Quadratic Function

Graph of the function `y=ax^2+bx+c`, where `a!=0` is parabola. To draw it three methods are used, that will be illustrated in following example.

Example . Draw graph of the function `y=-0.5x^2-x+4`.

First way: finding of coordinates `(x_0,y_0)` of vertex of parabola using following formulas: `x_0=-b/(2a)`; `y_0=(4ac-b^2)/(4a)`.

Drawing Graph of the Function y=f(kx)

Task 1. Draw graph of the function `y=f(kx)`, where `k>0,k!=1`, knowing graph of the function `y=f(x)`.

Let `y_0=f(x_0)`. Now answer the following question: what value of argument `x` should we take, so function `y=f(kx)` will take value `y_0`? Clearly this value should satisfy the following condition: `kx=x_0` or `x=(x_0)/k`. Therefore, point `(x_0;y_0)` that lies on graph of the given function `y=f(x)` is trasformed into point `((x_0)/k;y_0)` that lies on the graph of the function `y=f(kx)`. This transformation is called compressing of graph `y=f(x)` with coeffcient k to y-axis (if 0<k<1 then, in fact, we stretch from y-axis with coeffcient `1/k`).

Stretching and Compressing Graphs of Trigonometric Functions

Here we will talk about graphing functions of the form `y=msin(kx)`, `y=mcos(kx)`, `y=mtan(kx)`, `y=mcot(kx)`.

Example. Draw graph of the function `y=-3cos(2x)`.

Let's draw one half-wave of the graph `y=cos(x)`. We first compress it with coeffcient 2 to y-axis and obtain graph of the function. Now, we stretch graph of the function `y=cos(2x)` with coeffcient 3 from x-axis and reflect result about x-axis. In result we will obtain one half wave of the function `y=-3cos(2x)`. On a figure ileft graph is graph of the half-wave and right graph is all graph.

Graph of the Harmonic Oscillation `y=Asin(omega x+alpha)`

Trigonometric functions are used to describe oscillatory processes (for example, oscillation of pendulum). One of the most important formulas that describes such processes is `y=Asin(omega x+alpha)`, which is called formula of harmonic (or sinusoidal) oscillations. `A` is called amplitude of oscillation. `omega` is called frequency of oscillation. The bigger `omega` the more oscillations per unit of time. `alpha` is called starting phase of oscillation.

Definition of Trigonometric Expression

Expression, in which variable is written under signs of trigonometric functions is called trigonometric.

For example, `sin^4(x)+cos^4(x)`, `(tan(x/2))/(1+cot(x))` are trigonometric expressions. To transform trigonometric expressions properties of trigonometric functions and trigonometric formulas are used.

Formulas for Adding and Subtracting Arguments

For any real numbers `alpha` and `beta` following formulas are true:

  1. `cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)`,
  2. `cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta)`,
  3. `sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)`,
  4. `sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)`,
  5. `tan(alpha+beta)=(tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))`,
  6. `tan(alpha-beta)=(tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))`.

These formulas are called formulas for adding and subtracting arguments.

Reduction Formulas

Reduction formulas are formulas that allow to reduce function with argument of the form `(pi n)/2+-alpha,n in ZZ` to function with argument `alpha`.

Example. Find `sin(pi/2+alpha)`.

We have that `sin(pi/2+alpha)=sin(pi/2)cos(alpha)+cos(pi/2)sin(alpha)=1*cos(alpha)+0*sin(alpha)=cos(alpha)`.

The Simplest Transformations of Arithmetic Root (Radical)

When we transform the arithmetic root we should use their properties.

Example 1. Simplify the following: `sqrt(45a^5)`.

Using property 1, we will obtain `sqrt(45a^5)=sqrt(9a^4*5a)=sqrt(9)*sqrt(a^4)*sqrt(5a)=3a^2sqrt(5a)`.

Relation between Trigonometric Functions of Same Argument

We already know that `cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta)`.

If we take `alpha=t,beta=t` then `cos(t-t)=cos(t)cos(t)+sin(t)sin(t)` or `cos(0)=cos^2(t)+sin^2(t)`.

This gives us very important identity that connects sine and cosine (it is also called main trigonometric identity): `cos^2(t)+sin^2(t)=1`.

Double Angle Formulas

We already know following formulas:

  1. `sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)`,
  2. `cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)`,
  3. `tan(alpha+beta)=(tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))`.

If we take `alpha=t,beta=t` then we will obtain following three formulas:

Power-Reduction Formulas

We already know that `cos^2(t)+sin^2(t)=1` and `cos^2(t)-sin^2(t)=cos(2t)`.

Adding these formulas gives that `2cos^2(t)=1+cos(2t)` or `color(blue)(cos^2(t)=(1+cos(2t))/2)`.

Subtracting second formula from first gives that `2sin^2(t)=1-cos(2t)` or `color(green)(sin^2(t)=(1-cos(2t))/2)`.

Expressions for `sin(t),cos(t),tan(t)` through `tan(t/2)`

Using identities `sin(t)=2sin(t/2)cos(t/2)` and `cos^2(t/2)+sin^2(t/2)=1` we can write that

`sin(t)=(sin(t))/1=(2sin(t/2)cos(t/2))/(cos^2(t/2)+sin^2(t/2))`.

Dividing both numerator and denominator by `cos^2(t/2)` will give

Converting Sum of Trigonometric Functions into Product

To convert sum of trigonometric functions into sum following formulas are used:

  1. `sin(alpha)+sin(beta)=2sin((alpha+beta)/2)cos((alpha-beta)/2)`,
  2. `sin(alpha)-sin(beta)=2sin((alpha-beta)/2)cos((alpha+beta)/2)`,
  3. `cos(alpha)+cos(beta)=2cos((alpha+beta)/2)cos((alpha-beta)/2)`,
  4. `cos(alpha)-cos(beta)=-2sin((alpha+beta)/2)sin((alpha-beta)/2)`,
  5. `tan(alpha)+tan(beta)=(sin(alpha+beta))/(cos(alpha)cos(beta))`,
  6. `tan(alpha)-tan(beta)=(sin(alpha-beta))/(cos(alpha)cos(beta))`.

Formulas (5) and (6) are valid when `alpha` and `beta` not equal `pi/2+pi k, k in ZZ`.

Converting Product of Trigonometric Functions into Sum

To convert product of trigonometric functions into sum following formulas are used:

  1. `sin(alpha)cos(beta)=(sin(alpha-beta)+sin(alpha+beta))/2`,
  2. `sin(alpha)sin(beta)=(cos(alpha-beta)-cos(alpha+beta))/2`,
  3. `cos(alpha)cos(beta)=(cos(alpha-beta)+cos(alpha+beta))/2`.

Example. Convert into sum `sin(43^0)cos(19^0)`.

Converting Expression `acos(t)+bsin(t)` into the Form `Asin(t+alpha)`

Any expression of the form `acos(t)+bsin(t)` can be written in the form `Asin(t+alpha)`.

To do this let's factor out `sqrt(a^2+b^2)`:

`acos(t)+bsin(t)=sqrt(a^2+b^2)(a/(sqrt(a^2+b^2)) cos(t)+b/(sqrt(a^2+b^2))sin(t))`.

Transforming Expressions that contain Inverse trigonometric Functions

To transform expressions that contain inverse trigonometric function, we use definitions of these functions and trigonometric formulas.

Example 1. Simplify `sin(arccos(x))`, where `-1<=x<=1`.

The Identity `sqrt(a^2)=|a|`

Simplify the following expression `sqrt(a^2)`. There are two possible cases: `a>=0` or `a<0`. If `a>=0`, then `sqrt(a^2)=a`; if `a<0`, then `sqrt(a^2)=-a`. ` `

So, `sqrt(a^2)={(a if x>=0),(-a if x<0):}`.

Notion of Transcedental Expression

Expression is called transcedental if it contains variable under transcedental function, i.e. exponential, logarithmic, trigonometric or inverse trigonometric (see note functions y=arcsin(x), y=arccos(x), y=arctan(x)).

Definition of Logarithm of Positive Number with Given Base

Logarithm of positive number `x` with base `a` (`a>0,a!=1`) is called such exponent to which we need to raise number `a` to obtain `x`: `color(blue)(a^(log_a(x))=x)`.

Equality `log_a(x)=y` means that `a^y=x`.

Transformation of Irrational Expressions

For transformation of irrational expressions we use the properties of radicals and the properties of powers with rational exponents.

Example. Simlify the following expression:`f(x)=((root(4)(x^3)-root(4)(x))/(1-sqrt(x))+(1+sqrt(x))/root(4)(x))^2*(X^0+2/sqrt(x)+x^(-1))^(-1/2)`.

Properties of Logaritms

Properties of logarithms are following:

  1. If `x_1>0` and `x_2>0` then `color(red)(log_a(x_1x_2)=log_a(x_1)+log_a(x_2))` (logarithm of product equals sum of logarithms of factors). This property is true for any number of factors.

    For example, `log_3(60)=log_3(3*20)=log_3(3)+log_3(20)=1+log_3(20)=1+log_3(4*5)=1+log_3(4)+log_3(5)`.

Taking Logarithm and Exponentiating

If some expression A is formed from positive numbers with help of operations of multiplication, division and raising to power, then, using properties of logarithms, we can express `log_a(A)` in terms of logarithms of numbers that expression A contains. Such transformation is called taking logarithm.

Decimal Logarithm. Characteristic and Mantissa of Decimal Logarithm

If base of logarithm equasls 10, then logarithm is called decimal. Instead of record `log_(10)(x)` record `lg(x)` is used.

In particular, for decimal logarithms we have that `color(green)(10^(lg(a))=a)` and `color(blue)(lg(10^n)=n)`.