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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.
trigfunctions
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.
trigfunctions2
sine(s) = sin(s) = y. |
cosine(s) = cos(s) = x. |
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.
trigfunctions3
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.
The quadrantal angles have the values as shown in the chart below.
A reference angle is the positive acute angle created by the terminal side of an angle in standard position and the x-axis. A reference angle, therefore, is a first quadrant angle. Let β be the reference angle of θ. The value of a trigonometric function at θ is equal to the value of that function at β -- unless there is a variation in sign. The sign difference depends on which quadrant θ is in. An understanding of reference angles simplifies the evaluation of trigonometric functions of large angles.
It will be useful to memorize the values of the trigonometric functions at a few common angles besides the quadrantal angles: θ = 30o, 45o, 60circ. Use the values below to find the values of cosecant, secant, and cotangent at these angles.
values
sin(30o) = . |
cos(30o) = frac2. |
tan(30o) = . |
sin(45o) = . |
cos(45o) = frac1. |
tan(45o) = 1. |
sin(60o) = |
cos(60o) = frac12 |
tan(60o) = |
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