Trigonometric functions

Trigonometric functions are functions related to an angle. There are a total of 6 trigonometric functions, 3 of which, Sine, Cosine, and Tangent, are most commonly used, while the other 3, Secant, Cosecant, and Cotangent, are used less often.

Trigonometric function definitions

Consider an angle θ as one angle in a right triangle.

The following are the definitions of the trigonometric functions based on the right triangle above. In mathematics, these functions are often written in their abbreviated forms.

sine:
cosine:
tangent:
cosecant:
secant:
cotangent:

Example:

Given a=1, b=√3, and c=2, what are the values of sin(θ), cos(θ), and tan(θ)?

Unit-circle definitions

Trigonometric functions can also be defined using the coordinate values of a unit circle (a circle with radius one).

In a plane with a unit circle centered at the origin of a coordinate system, a ray from the origin forms an angle θ with respect to the x-axis. The ray intersects the unit circle at point A and intersects the line x=1 at point B. The coordinate values of A and B are (xA, yA) and (xB, yB).

sin(θ) = yA
cos(θ) = xA
tan(θ) = yB

For simplicity, only the 3 major trigonometric functions are drawn and defined above. The other 3 functions can be defined on the unit circle in a similar way.

One major limitation of defining trigonometric functions using right triangles is that the angles are restricted to being between 0° and 90°. Trigonometric functions defined using the unit circle do not have such limitations, and the definitions can thus be used with any angle.

Trigonometric function properties

The 6 trigonometric functions form many relationships. The following are some of the most important:

sin(-θ) = -sin(θ)
cos(-θ) = cos(θ)
tan(-θ) = -tan(θ)
cos(90° - θ) = sin(θ)
sin(90° - θ) = cos⁡(θ)
cos(90° + θ) = -sin(θ)
sin(90° + θ) = cos(θ)
sin(θ + 180°) = -sin(θ)
cos(θ + 180°) = -cos(θ)
sin(θ + n×360°) = sin(θ)
cos(θ + n×360°) = sin(θ)
sin2(θ) + cos2(θ) = 1
tan(θ + n×360°) = sin(θ)

In the equations above, n is an integer.

Also, aside from converting between different trigonometric functions, one of the main uses for the equations above is converting an angle of any value into an angle value between 0° and 90° to make the numbers easier to work with.

Example:

What is the value of sin(-750°)?

sin(-750°)= sin(2×360° - 750°)
 = sin(-30°)
 = -sin⁡(30°)
 = -0.5

Trigonometric function values

The values of trigonometric functions can be found through the coordinate values of the intersections on a unit circle. The trigonometric functions of many angles also have algebraic values. The following are a list of the algebraic values of some of the most commonly used angles.

The sign of trigonometric functions vary in different quadrants. The following are the signs of the 3 main trigonometric functions in each quadrant.

Inverse trigonometric functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions:

Arcsin, arccos, and arctan are the inverse functions of sine, cosine, and tangent respectively. The computation of inverse trigonometric functions results in an angle value.

Example:

Given sin(30°) = 0.5, what is arcsin(0.5)?

arcsin(0.5) = 30°

note: this only provides the results in the first quadrant. Since arcsin is a periodic function, arcsin(0.5) is also equal to 150°, 30°+n×360°, or 150°+n×360°, where n is an integer.