Chapter 5Trigonometric Functions

Section 5.3Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle

Consider angle q in the figure below, which is in standard position with a point P(x, y) on the terminal side of the angle. We define the trigonometric functions based upon this figure.

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle

The value of a trigonometric function is independent of the point chosen on the terminal side of the angle. Consider the figure below. The right triangles formed are similar triangles so the ratios of corresponding sides are equal.

Trigonometric Functions of Any Angle

Any point in the coordinate plane can determine an angle in standard position. Consider the figure below:

Example 1

Find the value of each of the six trigonometric functions of an angle q in standard position whose terminal side contains the point P(-3, -2).

Quadrantal Angles

Recall that a quadrantal angle is an angle who terminal side coincides with the x- or y-axis. Identify the point and then apply the appropriate trigonometric definition to find the value of the quadrantal function.

Quadrantal Angles

Signs of Trigonometric Functions

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies.

Example 2

Given that tan q = - , and sin q < 0, find cos q and csc .q

The Reference Angle

Given angle q in standard position, its reference angle q’ is the smallest positive angle formed by the terminal side of angle q and the x-axis.

Example 3

For each of the following, sketch the given angle q (in standard position) and its reference angle q’. Then determine the measure of q’.

.a q = 1200

.b q = 3450

.c q = 9240

.d q = p

.e q = -4 .f q = 17

Reference Angle Theorem

To evaluate the sin q, determine sin q’. Then use either sin q’ or its opposite as the answer, depending on which has the correct sign.

Example 4

Evaluate each function.

a. sin 2100

b. cos 4050

c. tan 3000

Assignment

Section 5.3 Worksheet