Trigonometry - sin, cos, tan, cot

Take an x-axis and an y-axis (orthonormal) and let O be the origin. A circle centered in O and with radius = 1 is known as trigonometric circle or unit circle.

If P is a point from the circle and t is the angle between PO and x then:

the x-coordinate of P is called the cosine of t. We write cos(t);
the y-coordinate of P is called the sine of t. We write sin(t);
the number sin(t)/cos(t) is called the tangent of t. We write tan(t);
the number cos(t)/sin(t) is called the cotangent of t. We write cot(t).

The sine function

sin : R -> R
All trygonometric functions are periodic. The period of sin is 2π.
The range of the function is [-1,1].

The cosine function

cos : R -> R
The period of sin is 2π.
The range of the function is [-1,1].

The tangent function

tan : R -> R
Now, the period is π and the images are not defined in x = (π/2) + kπ, k=0,1,2,...
The range or image is R.

The cotangent function

cot : R -> R
The period is π and that the images are not defined in x = kπ, k=0,1,2,...
The range or image is R.

Trigonometric formulas

With t radians corresponds exactly one point P(cos(t),sin(t)) on the unit circle. The square of the distance [OP] = 1. Calculating this distance with the coordinates of P we have for each t:

cos²(t) + sin²(t) = 1

If t + t' = 180° then: