# Trigonometric Functions

Consider 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].

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.

Domain 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.