﻿ ﻿IDENTITIES, EQUATIONS, AND INEQUALITIES - Trigonometric Functions and Their Inverses - Functions - REVIEW OF MAJOR TOPICS - SAT SUBJECT TEST MATH LEVEL 2

## PART 2 ## REVIEW OF MAJOR TOPICS ## CHAPTER 1Functions

### 1.3 Trigonometric Functions and Their Inverses ### IDENTITIES, EQUATIONS, AND INEQUALITIES

There are a few trigonometric identities you must know for the Mathematics Level 2 Subject Test.

Reciprocal Identities recognize the definitional relationships: Cofunction Identities were discussed earlier. Using radian measure:   Pythagorean Identities Double Angle Formulas EXAMPLES

1. Given cos and , find Since sin 2 = 2(sin )(cos ), you need to determine the value of sin . From the figure below, you can see that sin . Therefore, sin . 2. If cos 23° = z, find the value of cos 46° in terms of z.

Since 46 = 2(23), a double angle formula can be used: cos 2A = 2 cos2 A – 1. Substituting 23° for A, cos 46° = cos 2(23°) = 2 cos2 23° – 1 = 2(cos 23°)2 – 1 = 2z 2 – 1.

3. If sin x = A, find cos 2x in terms of A.

Using the identity cos 2x = 1 – sin2 x, you get cos 2x = 1 – A2.

You may be expected to solve trigonometric equations on the Math Level 2 Subject Test by using your graphing calculator and getting answers that are decimal approximations. To solve any equation, enter each side of the equation into a function (Yn), graph both functions, and find the point(s) of intersection on the indicated domain by choosing an appropriate window.

4. Solve 2 sin x + cos 2x = 2 sin2 x – 1 for 0 x 2 .

Enter 2 sin x + cos 2x into Y1 and 2 sin2 x – 1 into Y2. Set Xmin = 0, Xmax = 2 , Ymin = –4, and Ymax = 4. Solutions (x-coordinates of intersection points) are 1.57, 3.67, and 5.76.

5. Find values of x on the interval [0, ] for which cos x < sin 2x.

Enter each side of the inequality into a function, graph both, and find the values of x where the graph of cos x lies beneath the graph of sin 2x: 0.52 < x < 1.57 or x > 2.62.

EXERCISES

1. If sin and cos , find the value of sin 2x.

(A) – (B) – (C) (D) (E) 2. If tan A = cot B, then

(A) A = B

(B) A = 90° + B

(C) B = 90° + A

(D) A + B = 90°

(E) A + B = 180°

3. If cos , find cos 2x.

(A) –0.87

(B) –0.25

(C) 0

(D) 0.5

(E) 0.75

4. If sin 37° = z, express sin 74° in terms of z.

(A) (B) 2z 2 + 1

(C) 2z

(D) 2z 2 – 1

(E) 5. If sin x = –0.6427, what is csc x?

(A) –1.64

(B) –1.56

(C) 0.64

(D) 1.56

(E) 1.70

6. For what value(s) of x, 0 < x < , is sin x < cos x?

(A) x < 0.79

(B) x < 0.52

(C) 0.52 < x < 0.79

(D) x > 0.52

(E) x > 0.79

7. What is the range of the function f(x) = 5 – 6sin ( x + 1)?

(A) [–6,6]

(B) [–5,5]

(C) [–1,1]

(D) [–1,11]

(E) [–11,1]

﻿