Precalculus: Trigonometric Functions: Trigonometric Functions

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.

Figure %: The signs of the trigonometric functions in the quadrants

The quadrantal angles have the values as shown in the chart below.

Figure %: The values of the trigonometric functions of quadrantal angles

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.

Figure %: In each drawing, β is the reference angle for θ.

Figure %: How to calculate the reference angle β for any angle θ between 0 and 2Π radians.

It will be useful to memorize the values of the trigonometric functions at a few common angles besides the quadrantal angles: θ = 30^o, 45^o, 60^circ. Use the values below to find the values of cosecant, secant, and cotangent at these angles.

values

sin(30^o) =