﻿ ﻿SPECIAL ANGLES - Trigonometric Functions and Their Inverses - Functions - REVIEW OF MAJOR TOPICS - SAT SUBJECT TEST MATH LEVEL 2

## CHAPTER 1Functions

### SPECIAL ANGLES

When you use a calculator to evaluate most trig values, you will get a decimal approximation. You can use your knowledge of the definitions of the trigonometric functions, reference angles, and the ratios of the sides of the 45°-45°-90° triangle and the 30°-60°-90° triangle (“special” triangles) to get exact trig values for “special” angles: multiples of 30° , 45° , 60° .

The ratios of the sides of the two special triangles are shown in the figure below.

45° -45° -90° Triangle

30°- 60° -90° Triangle

To illustrate how this can be done, suppose you want to find the trig values of 120° First sketch the following graph.

The graph shows the angle in standard position, the reference angle 60°, and the (signed) side length ratios for the 30°-60°-90° triangle. You can now use the definitions of the trig functions to find the trig values:

Values can be checked by comparing the decimal approximation the calculator provides for the trig function with the decimal approximation obtained by entering the exact value in a calculator. In this example, sin 120° 0.866 and .

You can also readily obtain trig values of the quadrantal angles—multiples of 90° . The terminal sides of these angles are the x- and y-axes. In these cases, you don”t have a triangle at all; instead, either x or y equals 1 or –1, the other coordinate equals zero, and r equals 1. To illustrate how to use this method to evaluate the trig values of 270°, first draw the figure below.

The figure indicates x = 0 and y = –1 (r = 1). Therefore,

which is undefined

, which is undefined

EXERCISES

1. The exact value of tan (–60°) is

(A) –

(B) –1

(C)

(D)

(E)

2. The exact value of cos

(A) –1

(B)

(C)

(D)

(E) 0

3. Csc 540° is

(A) 0

(B) –

(C)

(D) –1

(E) undefined

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