Free Printable Chapter 5: Trigonometric Identities & Equations

1.

Simplify cos x tan x.

a)

csc x

b)

sin x

c)

1

d)

cos x

2.

Simplify cos x + cos x tan² x

a)

sec x

b)

1 + tan x

c)

csc x

d)

tan x

3.

If csc x = 2, find sin x .

a)

$\sqrt{2}$

b)

$\frac{\sqrt{3}}{2}$

c)

1/2

d)

2

4.

Determine the exact vaue of cos x, given sin x = 4/5 and $0\degree<x<90\degree$ .

a)

5/3

b)

3/5

c)

4/5

d)

5/4

5.

Complete the identity $\frac{\sin x\cos x}{1-\cos^2x}=_{ }$ ________ .

a)

cos x

b)

tan x

c)

cot x

d)

sin x

6.

Complete the identity of $\frac{\cot\ x+\tan x}{\sec^2x}=$ _____________.

a)

cos x

b)

cot x

c)

tan x

d)

sin x

7.

Complete the identity csc x (cos x - sec x) = ___________.

a)

-tan x

b)

tan x

c)

cot x

d)

-cot x

8.

Complete the identity $\frac{\sec^2x\sin^2x}{\sec x-\cos x}=\$ ___________.

a)

csc x

b)

sec x

c)

tan x

d)

cot x

9.

Solve $\sin x\cos x-\frac{\sqrt{3}}{2}\cos x\ =\ 0$

for principal values of x. Express solutions in degrees.

a)

30 degrees, 60 degrees

b)

150 degrees, 90 degrees

c)

30 degrees, 90 degrees

d)

60 degrees, 90 degrees

10.

Solve $\sin^2x-\sin x+1=\cos^2x\ for\ 0\le x<2\pi$ .

a)

$0,\ \frac{\pi}{6},\ \frac{5\pi}{6}$

b)

$0,\ \frac{3\pi}{4},\ \frac{5\pi}{6},\ \pi$

c)

$0,\ \frac{\pi}{6},\ \frac{5\pi}{6},\ \pi$

d)

$0,\ \frac{\pi}{6},\ \frac{\pi}{3},\ \pi$

11.

Solve $4\sin^2x+3=4$ for principal values of x. Express solutions in radians.

a)

$\frac{\pi}{6},\ \frac{5\pi}{6}$

b)

$\frac{\pi}{3},\ \frac{2\pi}{3}$

c)

$-\frac{\pi}{6},\ \frac{\pi}{6}$

d)

$-\frac{\pi}{3},\ \frac{\pi}{3}$

12.

Solve $\sqrt{2}\cos x-1\ =\ 0$ for all real values of x.

a)

$\frac{\pi}{4}+2\pi k,\ \frac{7\pi}{4}+2\pi k$

b)

$\frac{\pi}{6}+2\pi k,\ \frac{5\pi}{6}+2\pi k$

c)

$\frac{\pi}{4}+\pi k$

d)

$\frac{3\pi}{4}+2\pi k,\ \frac{5\pi}{4}+2\pi k$

13.

Use the sum or difference identity for sine to find the exact value of sin $375\degree$ .

a)

$\frac{\sqrt{3}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

14.

Find the value of $\sin\left(x+y\right)\ if\ 0<x<\frac{\pi}{2},\ 0<y<\frac{\pi}{2},\ \sin x=\frac{2}{3},\ and\ \sin y=\frac{1}{2}$

a)

$\frac{2\sqrt{3}+\sqrt{5}}{6}$

b)

$\frac{\sqrt{6}+\sqrt{5}}{3}$

c)

$\frac{\sqrt{6}+\sqrt{5}}{3}$

d)

$\frac{\sqrt{6}-\sqrt{5}}{6}$

15.

If x and y are acute angles such that sin x = 1/3 and sin y = 1/4, what is the value of cos (x - y)?

a)

$\frac{-1+2\sqrt{30}}{12}$

b)

$\frac{1+2\sqrt{30}}{12}$

c)

$\frac{1-2\sqrt{30}}{12}$

d)

$\frac{-1-2\sqrt{30}}{12}$

16.

Complete the identity $\cos\left(\frac{3\pi}{2}+A\right)$ = ______________.

a)

sin A

b)

-cos A

c)

-sin A

d)

cos A

17.

If sin x = 1/3 and x has its terminal side in the first quadrant, find the exact value of sin 2x.

a)

$\frac{2\sqrt{2}}{3}$

b)

$\frac{4\sqrt{2}}{3}$

c)

$\frac{4\sqrt{2}}{9}$

d)

$\frac{2}{3}$

18.

Use a half-angle identity to find the exact value of $\sin\ 67.5\degree$ .

a)

$\frac{\sqrt{2+\sqrt{2}}}{2}$

b)

$\frac{\sqrt{1+\sqrt{2}}}{2}$

c)

$\frac{\sqrt{2-\sqrt{2}}}{2}$

d)

$\frac{\sqrt{1-\sqrt{2}}}{2}$

19.

$Find\ \sin2x\ if\ \sin x=\frac{1}{5}and\ 0\degree<x<90\degree$

.

a)

3/5

b)

$\frac{4\sqrt{6}}{25}$

c)

4/5

d)

$\frac{2\sqrt{6}}{5}$

20.

Complete the identity $\frac{\sin2x}{1+\cos2x}=\$ _____________.

a)

cot x

b)

cos x

c)

tan x

d)

sin x

Chapter 5: Trigonometric Identities & Equations

Total questions: 20

Worksheet time: 40mins

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