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Amy Ferguson Moncure

PreCalc 2 · Analytic Trigonometry

Sum and Difference Formulas

Write the angle as a sum or a difference of angles whose sine, cosine, or tangent you know. Then use the formula.

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Key Ideas Easy Medium
What to Remember

Sum formulas

sin(A + B) = sin A cos B + cos A sin B

cos(A + B) = cos A cos B − sin A sin B

tan(A + B) = tan A + tan B1 − tan A · tan B

Difference formulas

sin(A − B) = sin A cos B − cos A sin B

cos(A − B) = cos A cos B + sin A sin B

tan(A − B) = tan A − tan B1 + tan A · tan B

Exact values

Split the angle into 30°, 45°, or 60°.

75° = 45° + 30°

15° = 45° − 30°

105° = 60° + 45°

A given sine or cosine

Use sin² + cos² = 1 to find the other ratio.

The quadrant tells you the sign.

Then substitute into the sum or difference formula.

Level 1

Easy

Split the angle into 30°, 45°, or 60°.

Example
Find the exact value of sin 75°.

sin(A + B) = sin A cos B + cos A sin B

A = 45°

B = 30°

sin 45° · cos 30° + cos 45° · sin 30°

sin 45° = square root of 22

cos 30° = square root of 32

cos 45° = square root of 22

sin 30° = 12

sin 45° · cos 30°

square root of 64

cos 45° · sin 30°

square root of 24

square root of 6 + square root of 24

Answer: square root of 6 + square root of 24

1
Find the exact value of cos 75°.
Show Answer

cos(A + B) = cos A cos B − sin A sin B

A = 45°

B = 30°

cos 45° · cos 30° − sin 45° · sin 30°

cos 45° = square root of 22

cos 30° = square root of 32

sin 45° = square root of 22

sin 30° = 12

cos 45° · cos 30°

square root of 64

sin 45° · sin 30°

square root of 24

square root of 6 − square root of 24

Answer: square root of 6 − square root of 24

2
Find the exact value of sin 15°.
Show Answer

sin(A − B) = sin A cos B − cos A sin B

A = 45°

B = 30°

sin 45° · cos 30° − cos 45° · sin 30°

sin 45° = square root of 22

cos 30° = square root of 32

cos 45° = square root of 22

sin 30° = 12

sin 45° · cos 30°

square root of 64

cos 45° · sin 30°

square root of 24

square root of 6 − square root of 24

Answer: square root of 6 − square root of 24

3
Find the exact value of cos 15°.
Show Answer

cos(A − B) = cos A cos B + sin A sin B

A = 45°

B = 30°

cos 45° · cos 30° + sin 45° · sin 30°

cos 45° = square root of 22

cos 30° = square root of 32

sin 45° = square root of 22

sin 30° = 12

cos 45° · cos 30°

square root of 64

sin 45° · sin 30°

square root of 24

square root of 6 + square root of 24

Answer: square root of 6 + square root of 24

4
Find the exact value of tan 75°.
Show Answer

tan(A + B) = tan A + tan B1 − tan A · tan B

A = 45°

B = 30°

tan 45° = 1

tan 30° = square root of 33

tan 45° + tan 30°

3 + square root of 33

tan 45° · tan 30°

square root of 33

1 − tan A · tan B

3 − square root of 33

2 + square root of 3

Answer: 2 + square root of 3

5
Find the exact value of tan 15°.
Show Answer

tan(A − B) = tan A − tan B1 + tan A · tan B

A = 45°

B = 30°

tan 45° = 1

tan 30° = square root of 33

tan 45° − tan 30°

3 − square root of 33

tan 45° · tan 30°

square root of 33

1 + tan A · tan B

3 + square root of 33

2 − square root of 3

Answer: 2 − square root of 3

6
Find the exact value of sin 105°.
Show Answer

sin(A + B) = sin A cos B + cos A sin B

A = 60°

B = 45°

sin 60° · cos 45° + cos 60° · sin 45°

sin 60° = square root of 32

cos 45° = square root of 22

cos 60° = 12

sin 45° = square root of 22

sin 60° · cos 45°

square root of 64

cos 60° · sin 45°

square root of 24

square root of 6 + square root of 24

Answer: square root of 6 + square root of 24

7
Find the exact value of cos 105°.
Show Answer

cos(A + B) = cos A cos B − sin A sin B

A = 60°

B = 45°

cos 60° · cos 45° − sin 60° · sin 45°

cos 60° = 12

cos 45° = square root of 22

sin 60° = square root of 32

sin 45° = square root of 22

cos 60° · cos 45°

square root of 24

sin 60° · sin 45°

square root of 64

square root of 2 − square root of 64

Answer: square root of 2 − square root of 64

8
Find the exact value of tan 105°.
Show Answer

tan(A + B) = tan A + tan B1 − tan A · tan B

A = 60°

B = 45°

tan 60° = square root of 3

tan 45° = 1

tan 60° + tan 45°

1 + square root of 3

tan 60° · tan 45°

square root of 3

1 − tan A · tan B

1 − square root of 3

−(2 + square root of 3)

Answer: −(2 + square root of 3)

9
Find the exact value of sin 11π12.
Show Answer

sin(A + B) = sin A cos B + cos A sin B

A = 2π3

B = π4

sin 2π3 · cos π4 + cos 2π3 · sin π4

sin 2π3 = square root of 32

cos π4 = square root of 22

cos 2π3 = −12

sin π4 = square root of 22

sin 2π3 · cos π4

square root of 64

cos 2π3 · sin π4

−square root of 24

square root of 6 − square root of 24

Answer: square root of 6 − square root of 24

10
Find the exact value of cos 11π12.
Show Answer

cos(A + B) = cos A cos B − sin A sin B

A = 2π3

B = π4

cos 2π3 · cos π4 − sin 2π3 · sin π4

cos 2π3 = −12

cos π4 = square root of 22

sin 2π3 = square root of 32

sin π4 = square root of 22

cos 2π3 · cos π4

−square root of 24

sin 2π3 · sin π4

square root of 64

−square root of 6 + square root of 24

Answer: −square root of 6 + square root of 24

Level 2

Medium

You are given a ratio and a quadrant for each angle.

Example
Find the exact value of sin(A + B). sin A = 35 and A is in Quadrant I. cos B = 513 and B is in Quadrant I.

sin(A + B) = sin A cos B + cos A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 35

sin² A = 925

1 − 925

2525 − 925

1625

cos² A = 1625

A is in Quadrant I, so cosine is positive.

cos A = 45

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = 513

cos² B = 25169

1 − 25169

169169 − 25169

144169

sin² B = 144169

B is in Quadrant I, so sine is positive.

sin B = 1213

sin A · cos B + cos A · sin B

sin A · cos B

35 · 513

1565

313

cos A · sin B

45 · 1213

4865

313 + 4865

1565 + 4865

6365

Answer: 6365

1
Find the exact value of sin(A + B). sin A = 45 and A is in Quadrant I. cos B = 1213 and B is in Quadrant I.
Show Answer

sin(A + B) = sin A cos B + cos A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 45

sin² A = 1625

1 − 1625

2525 − 1625

925

cos² A = 925

A is in Quadrant I, so cosine is positive.

cos A = 35

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = 1213

cos² B = 144169

1 − 144169

169169 − 144169

25169

sin² B = 25169

B is in Quadrant I, so sine is positive.

sin B = 513

sin A · cos B + cos A · sin B

sin A · cos B

45 · 1213

4865

cos A · sin B

35 · 513

1565

313

4865 + 313

4865 + 1565

6365

Answer: 6365

2
Find the exact value of cos(A + B). sin A = 513 and A is in Quadrant I. cos B = 35 and B is in Quadrant I.
Show Answer

cos(A + B) = cos A cos B − sin A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 513

sin² A = 25169

1 − 25169

169169 − 25169

144169

cos² A = 144169

A is in Quadrant I, so cosine is positive.

cos A = 1213

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = 35

cos² B = 925

1 − 925

2525 − 925

1625

sin² B = 1625

B is in Quadrant I, so sine is positive.

sin B = 45

cos A · cos B − sin A · sin B

cos A · cos B

1213 · 35

3665

sin A · sin B

513 · 45

2065

413

3665 − 413

3665 − 2065

1665

Answer: 1665

3
Find the exact value of sin(A − B). sin A = 817 and A is in Quadrant I. cos B = 725 and B is in Quadrant I.
Show Answer

sin(A − B) = sin A cos B − cos A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 817

sin² A = 64289

1 − 64289

289289 − 64289

225289

cos² A = 225289

A is in Quadrant I, so cosine is positive.

cos A = 1517

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = 725

cos² B = 49625

1 − 49625

625625 − 49625

576625

sin² B = 576625

B is in Quadrant I, so sine is positive.

sin B = 2425

sin A · cos B − cos A · sin B

sin A · cos B

817 · 725

56425

cos A · sin B

1517 · 2425

360425

7285

56425 − 7285

56425 − 360425

−304425

Answer: −304425

4
Find the exact value of cos(A − B). cos A = 817 and A is in Quadrant I. sin B = 35 and B is in Quadrant I.
Show Answer

cos(A − B) = cos A cos B + sin A sin B

sin² A + cos² A = 1

sin² A = 1 − cos² A

cos A = 817

cos² A = 64289

1 − 64289

289289 − 64289

225289

sin² A = 225289

A is in Quadrant I, so sine is positive.

sin A = 1517

sin² B + cos² B = 1

cos² B = 1 − sin² B

sin B = 35

sin² B = 925

1 − 925

2525 − 925

1625

cos² B = 1625

B is in Quadrant I, so cosine is positive.

cos B = 45

cos A · cos B + sin A · sin B

cos A · cos B

817 · 45

3285

sin A · sin B

1517 · 35

4585

917

3285 + 917

3285 + 4585

7785

Answer: 7785

5
Find the exact value of tan(A + B). tan A = 34. tan B = 512.
Show Answer

tan(A + B) = tan A + tan B1 − tan A · tan B

tan A = 34

tan B = 512

tan A + tan B

34 + 512

912 + 512

1412

76

tan A · tan B

34 · 512

1548

516

1 − tan A · tan B

1 − 516

1616 − 516

1116

761116

76 · 1611

5633

Answer: 5633

6
Find the exact value of sin(A + B). sin A = 513 and A is in Quadrant II. cos B = 45 and B is in Quadrant I.
Show Answer

sin(A + B) = sin A cos B + cos A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 513

sin² A = 25169

1 − 25169

169169 − 25169

144169

cos² A = 144169

A is in Quadrant II, so cosine is negative.

cos A = −1213

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = 45

cos² B = 1625

1 − 1625

2525 − 1625

925

sin² B = 925

B is in Quadrant I, so sine is positive.

sin B = 35

sin A · cos B + cos A · sin B

sin A · cos B

513 · 45

2065

413

cos A · sin B

(−1213) · 35

−3665

413 − 3665

2065 − 3665

−1665

Answer: −1665

7
Find the exact value of cos(A − B). cos A = −1517 and A is in Quadrant II. sin B = −725 and B is in Quadrant IV.
Show Answer

cos(A − B) = cos A cos B + sin A sin B

sin² A + cos² A = 1

sin² A = 1 − cos² A

cos A = −1517

cos² A = 225289

1 − 225289

289289 − 225289

64289

sin² A = 64289

A is in Quadrant II, so sine is positive.

sin A = 817

sin² B + cos² B = 1

cos² B = 1 − sin² B

sin B = −725

sin² B = 49625

1 − 49625

625625 − 49625

576625

cos² B = 576625

B is in Quadrant IV, so cosine is positive.

cos B = 2425

cos A · cos B + sin A · sin B

cos A · cos B

(−1517) · 2425

−360425

−7285

sin A · sin B

817 · (−725)

−56425

−7285 − 56425

−360425 − 56425

−416425

Answer: −416425

8
Find the exact value of tan(A − B). tan A = 2. tan B = −13.
Show Answer

tan(A − B) = tan A − tan B1 + tan A · tan B

tan A = 2

tan B = −13

tan A − tan B

2 + 13

63 + 13

73

tan A · tan B

2 · (−13)

−23

1 + tan A · tan B

1 − 23

33 − 23

13

7313

73 · 3

7

Answer: 7

9
Find the exact value of sin(A + B). sin A = 2029 and A is in Quadrant I. cos B = −35 and B is in Quadrant II.
Show Answer

sin(A + B) = sin A cos B + cos A sin B

sin² A + cos² A = 1

cos² A = 1 − sin² A

sin A = 2029

sin² A = 400841

1 − 400841

841841 − 400841

441841

cos² A = 441841

A is in Quadrant I, so cosine is positive.

cos A = 2129

sin² B + cos² B = 1

sin² B = 1 − cos² B

cos B = −35

cos² B = 925

1 − 925

2525 − 925

1625

sin² B = 1625

B is in Quadrant II, so sine is positive.

sin B = 45

sin A · cos B + cos A · sin B

sin A · cos B

2029 · (−35)

−60145

−1229

cos A · sin B

2129 · 45

84145

−1229 + 84145

−60145 + 84145

24145

Answer: 24145

10
Find the exact value of cos(A + B). cos A = −817 and A is in Quadrant III. sin B = −2029 and B is in Quadrant IV.
Show Answer

cos(A + B) = cos A cos B − sin A sin B

sin² A + cos² A = 1

sin² A = 1 − cos² A

cos A = −817

cos² A = 64289

1 − 64289

289289 − 64289

225289

sin² A = 225289

A is in Quadrant III, so sine is negative.

sin A = −1517

sin² B + cos² B = 1

cos² B = 1 − sin² B

sin B = −2029

sin² B = 400841

1 − 400841

841841 − 400841

441841

cos² B = 441841

B is in Quadrant IV, so cosine is positive.

cos B = 2129

cos A · cos B − sin A · sin B

cos A · cos B

(−817) · 2129

−168493

sin A · sin B

(−1517) · (−2029)

300493

−168493 − 300493

−468493

Answer: −468493

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