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

## CHAPTER 1Functions

### 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 2= 2 cos2 – 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 A, find cos 2in terms of A.

Using the identity cos 2= 1 – sin2 x, you get cos 2= 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 + cos 2= 2 sin2 – 1 for 0   2.

Enter 2 sin + cos 2into Y1 and 2 sin– 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 on the interval [0,for which cos < sin 2x.

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

EXERCISES

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

(A)  –

(B)  –

(C)

(D)

(E)

2.       If tan = cot B, then

(A)  B

(B)  = 90° + B

(C)  = 90° + A

(D)  = 90°

(E)  = 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 = –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 < , is sin < cos x?

(A)  < 0.79

(B)  < 0.52

(C)  0.52 < < 0.79

(D)  > 0.52

(E)  > 0.79

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

(A)  [–6,6]

(B)  [–5,5]

(C)  [–1,1]

(D)  [–1,11]

(E)  [–11,1]

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