Trigonometric Functions

definition of trigonometric functionsConsider unit circle centered at origin and point `P_0(1,0)`. If we begin to rotate point `P_0` around origin on angle `t` then we will obtain point `P_t`.

x-coordinate of this point is called cosine of number `t` and denoted by `cos(t)`, y-coordinate of this point is called sine of number `t` and denoted by `sin(t)`.

Tangent of number `t` is ratio of sine and cosine: `tan(t)=(sin(t))/(cos(t))`.

Cotangent of number `t` is ratio of cosine and sine: `cot(t)=(cos(t))/(sin(t))`.

Secant of number `t` is `sec(t)=1/(cos(t))`.

Cosecant of number `t` is `csc(t)=1/(sin(t))`.

When we talk about trigonometric functions we can use both radian and degree measure of angle `t`, but in calculus we almost always use radian measure (unless other stated).

To convert radian measure to degree and vice versa following formulas are used:

`1\ rad=(180^0)/pi~~57^0` and `1^0=pi/(180^0)\ rad~~0.017\ rad`.

So, `pi` is `180^0`, `2pi` is `360^0`, `pi/2` is `90^0` etc.

Domain of cosine and sine is `(-oo,oo)`, their range is `[-1,1]`.sine and cosine

These functions are periodic with main period `2pi`, i.e. `sin(x+2pi)=sin(x)` and `cos(x+2pi)=cos(x)` for all `x`.

Domain of tangent function is all `x` except those `x` where `cos(x)=0`.

Range of tangent function is `(-oo,oo)`.

Tangent is periodic function with period `pi`: `tan(x+pi)=tan(x)` for all `x`.

tangent and cotangent functionsDomain of cotangent function is all `x` except those `x` where `sin(x)=0`.

Range of cotangent function is `(-oo,oo)`. Cotangent is periodic function with period `pi`: `cot(x+pi)=cot(x)` for all `x`.

Following formulas hold for trigonometric functions. They will be used in further notes:

  1. `cos^2(x)+sin^2(x)=1` for all `x`.
  2. `1+tan^2(x)=sec^2(x)` for all `x`.
  3. `1+cot^2(x)=csc^2(x)` for all `x`.
  4. `sin(x+-y)=sin(x)cos(y)+-cos(x)sin(y)` for all `x,y`.
  5. `cos(x+y)=cos(x)cos(y)-sin(x)sin(y)` for all `x,y`.
  6. `cos(x-y)=cos(x)cos(y)+sin(x)sin(y)` for all `x,y`.
  7. `tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan(y))` for all `x,y`.
  8. `tan(x-y)=(tan(x)-tan(y))/(1+tan(x)tan(y))` for all `x,y`.

Trigonometric functions, because of periodicity, are widely used for modeling repetitive events: motion of pendulum, vibrating string, sound waves etc.