There are three basic ways to define the trigonometric functions.
Consider a point (x, y) on the terminal side of an angle θ in
standard position. It lies a distance d away from the origin.
|sine(θ) = sin(θ) = .||
|cosine(θ) = cos(θ) = .||
|tangent(θ) = tan(θ) = .||
|cosecant(θ) = csc(θ) = .||
|secant(θ) = sec(θ) = .||
|cotangent(θ) = cot(θ) = .||
The unit circle is the circle with equation x2 + y2 = 1. Let s be the
length of the arc with one endpoint at (1, 0) extending around the circle
counterclockwise with its other endpoint at (x, y). Note that s is both the
length of an arc as well as the measure in radians of the central angle that
intercepts that arc.
|tangent(s) = tan(s) = ;;x≠0.||
|cosecant(s) = csc(s) = ;;y≠0.||
|secant(s) = sec(s) = ;;x≠0.||
|cotangent(s) = cot(s) = ;;y≠0.||
Consider a right triangle with one acute angle θ in standard position.
Let the side opposite that angle be called the opposite side. Let the other leg
be called the adjacent side.
|sine(θ) = sin(θ) = ||
|cosine(θ) = cos(θ) = ||
|tangent(θ) = tan(θ) = ||
|cosecant(θ) = csc(θ) = ||
|secant(θ) = sec(θ) = ||
|cotangent(θ) = cot(θ) = ||
The trigonometric functions have different signs according to the quadrant
in which the angle's terminal side lies. Here is a chart to show these signs.
Figure %: The signs of the trigonometric functions in the quadrants
The quadrantal angles have the values as shown in the chart below.