## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 1

Functions

###

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 2*A *= 2 cos^{2} *A *– 1. Substituting 23° for *A*, cos 46° = cos 2(23°) = 2 cos^{2} 23° – 1 = 2(cos 23°)^{2} – 1 = 2*z* ^{2} – 1.

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

Using the identity cos 2*x *= 1 – sin^{2} *x*, you get cos 2*x *= 1 – *A*^{2}.

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 (Y_{n}), 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 sin^{2} x – 1 for 0**

*x***2**

**.**

Enter 2 sin *x *+ cos 2*x *into Y_{1} and 2 sin^{2 }*x *– 1 into Y_{2}. 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 2*x.*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 2*x*: 0.52 < *x *< 1.57 or *x *> 2.62.

**EXERCISES**

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

(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 2*x*.

(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) 2*z* ^{2} + 1

(C) 2*z*

(D) 2*z* ^{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]