﻿ Sum and Difference Identities Worksheet | Problems & Solutions

# Sum and Difference Identities Worksheet

Sum and Difference Identities Worksheet
• Page 1
1.
Find the value of cos 8° cos 4° + sin 8° sin 4°.
 a. 0.9781 b. 0.9975 c. 1

#### Solution:

cos 8° cos 4° + sin 8° sin 4°

= cos (8° - 4°)
[Use cos A cos B + sin A sin B = cos (A - B).]

= cos 4° = 0.9975

2.
sin 33° = ?
 a. 0.4675 b. 0.0769 c. 0.5444

#### Solution:

sin 33° = sin (5° + 28°)

= sin 5° cos 28° + cos 5° sin 28°
[Use sin (A + B) = sin A cos B + cos A sin B.]

= (0.0871)(0.8829) + (0.9961)(0.4694) = 0.5444

3.
cos 16° = ?
 a. 0.961 b. 0.9372 c. 0.0238

#### Solution:

cos 16° = cos (4° - 20°)

= cos 4° cos 20° + sin 4° sin 20°
[Use cos (A - B) = cos A cos B + sin A sin B.]

= (0.9975)(0.9396) + (0.0697)(0.3420) = 0.9610

4.
cos 28° = ?
 a. 0.0283 b. 0.8829 c. 0.9112

#### Solution:

cos 28° = cos (4° + 24°)

= cos 4° cos 24° - sin 4° sin 24°
[Use cos (A + B) = cos A cos B - sin A sin B.]

= (0.9975)(0.9135) - (0.0697)(0.4067) = 0.8829

5.
tan ($\frac{\mathrm{5\pi }}{\mathrm{12}}$) = ?
 a. $\sqrt{3}$ - 2 b. - 2 - $\sqrt{3}$ c. 2 + $\sqrt{3}$ d. 2 - $\sqrt{3}$

#### Solution:

tan (12 ) = tan (π4 + π6 )

= tanπ4 + tanπ61 - tanπ4tanπ6
[Use tan (A + B) = tan A + tan B1 - tan A tan B.]

= 1+331-33 = 3+33-3

= 3+33-3 · 3+33+3

= 9 + 63 + 39 - 3

= 12 + 636

= 2 + 3

6.
Find sin 9A cos 6A + cos 9A sin 6A .
 a. sin 3A b. cos 15A c. sin 15A d. cos 3A

#### Solution:

sin 9A cos 6A + cos 9A sin 6A

= sin (9A + 6A)
[Use sin A cos B + cos A sin B = sin (A + B).]

= sin 15A

7.
sin (5$\pi$ + 6$\theta$) = _________
 a. - cos 6$\theta$ b. sin 6$\theta$ c. cos 6$\theta$ d. - sin 6$\theta$

#### Solution:

sin (5π + 6θ) = sin 5π cos 6θ + cos 5π sin 6θ
[Use sin (A + B) = sin A cos B + cos A sin B.]

= (0)cos 6θ + (- 1)sin 6θ

= - sin 6θ

8.
Find sin 70°30′ cos 10°30′ - cos 70°30′sin 10°30′.
 a. $\frac{1}{2}$ b. sin 81° c. $\frac{\sqrt{3}}{2}$ d. cos 81°

#### Solution:

sin 70°30′ cos 10°30′ - cos 70°30′sin 10°30′

= sin (70°30′ - 10°30′)
[Use sin A cos B - cos A sin B = sin (A - B).]

= sin 60° = 32

9.
Evaluate: cos 5° cos 4° - sin 5° sin 4°
 a. 1 b. -1 c. 0.9876

#### Solution:

cos 5° cos 4° - sin 5° sin 4°
[Use cos A cos B - sin A sin B = cos (A + B).]

= cos (5° + 4°)

= cos 9° = 0.9876
[Check using calculator.]

10.
Find the value of sin 13° cos 9° + cos 13° sin 9°.
 a. 0.0697 b. 0.3746 c. 1

#### Solution:

sin 13° cos 9° + cos 13° sin 9°
[Use sin A cos B + cos A sin B = sin (A + B).]

= sin (13° + 9°)

= sin 22° = 0.3746
[Check using calculator.]