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Amy Ferguson Moncure

PreCalc 2 · Analytic Trigonometry

Power-Reducing Formulas

Use a power-reducing formula to find an exact value of a square of sine, cosine, or tangent.

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Key Ideas Easy Medium
What to Remember

Power-reducing formulas

sin² θ = 1 − cos 2θ2

cos² θ = 1 + cos 2θ2

tan² θ = 1 − cos 2θ1 + cos 2θ

A known double angle

15° uses cos 30°.

22.5° uses cos 45°.

75° uses cos 150°.

The answer is a square, so there is no ±.

A given cosine

Substitute cos 2θ into the formula.

The formula already gives the square, so the quadrant is not needed.

A harder square

For tan² θ, divide, then rationalize if a square root remains in the denominator.

sin⁴ θ means (sin² θ)². Find the square, then square that value.

Level 1

Easy

The double angle is one you know, or you are given cos 2θ.

Example
Find the exact value of sin² 15°.

sin² θ = 1 − cos 2θ2

θ = 15°

2θ = 30°

cos 30° = square root of 32

1 − cos 30°

1 − square root of 32

22 − square root of 32

2 − square root of 32

Divide by 2.

2 − square root of 322

2 − square root of 32 · 12

2 − square root of 34

Answer: 2 − square root of 34

1
Find the exact value of cos² 15°.
Show Answer

cos² θ = 1 + cos 2θ2

θ = 15°

2θ = 30°

cos 30° = square root of 32

1 + cos 30°

1 + square root of 32

22 + square root of 32

2 + square root of 32

Divide by 2.

2 + square root of 322

2 + square root of 32 · 12

2 + square root of 34

Answer: 2 + square root of 34

2
Find the exact value of sin² 22.5°.
Show Answer

sin² θ = 1 − cos 2θ2

θ = 22.5°

2θ = 45°

cos 45° = square root of 22

1 − cos 45°

1 − square root of 22

22 − square root of 22

2 − square root of 22

Divide by 2.

2 − square root of 222

2 − square root of 22 · 12

2 − square root of 24

Answer: 2 − square root of 24

3
Find the exact value of cos² 22.5°.
Show Answer

cos² θ = 1 + cos 2θ2

θ = 22.5°

2θ = 45°

cos 45° = square root of 22

1 + cos 45°

1 + square root of 22

22 + square root of 22

2 + square root of 22

Divide by 2.

2 + square root of 222

2 + square root of 22 · 12

2 + square root of 24

Answer: 2 + square root of 24

4
Find the exact value of sin² 75°.
Show Answer

sin² θ = 1 − cos 2θ2

θ = 75°

2θ = 150°

cos 150° = −square root of 32

1 − cos 150°

1 − (−square root of 32)

1 + square root of 32

22 + square root of 32

2 + square root of 32

Divide by 2.

2 + square root of 322

2 + square root of 32 · 12

2 + square root of 34

Answer: 2 + square root of 34

5
Find the exact value of cos² 75°.
Show Answer

cos² θ = 1 + cos 2θ2

θ = 75°

2θ = 150°

cos 150° = −square root of 32

1 + cos 150°

1 + (−square root of 32)

1 − square root of 32

22 − square root of 32

2 − square root of 32

Divide by 2.

2 − square root of 322

2 − square root of 32 · 12

2 − square root of 34

Answer: 2 − square root of 34

6
cos 2θ = 35. Find sin² θ.
Show Answer

sin² θ = 1 − cos 2θ2

cos 2θ = 35

1 − cos 2θ

1 − 35

55 − 35

25

Divide by 2.

252

25 · 12

210

15

Answer: 15

7
cos 2θ = 513. Find cos² θ.
Show Answer

cos² θ = 1 + cos 2θ2

cos 2θ = 513

1 + cos 2θ

1 + 513

1313 + 513

1813

Divide by 2.

18132

1813 · 12

1826

913

Answer: 913

8
cos 2θ = 725. Find sin² θ.
Show Answer

sin² θ = 1 − cos 2θ2

cos 2θ = 725

1 − cos 2θ

1 − 725

2525 − 725

1825

Divide by 2.

18252

1825 · 12

1850

925

Answer: 925

9
cos 2θ = 817. Find cos² θ.
Show Answer

cos² θ = 1 + cos 2θ2

cos 2θ = 817

1 + cos 2θ

1 + 817

1717 + 817

2517

Divide by 2.

25172

2517 · 12

2534

Answer: 2534

10
cos 2θ = 45. Find sin² θ.
Show Answer

sin² θ = 1 − cos 2θ2

cos 2θ = 45

1 − cos 2θ

1 − 45

55 − 45

15

Divide by 2.

152

15 · 12

110

Answer: 110

Level 2

Medium

Rationalize a tangent square, square again for a fourth power, or use a negative cosine.

Example
Find the exact value of tan² 15°.

tan² θ = 1 − cos 2θ1 + cos 2θ

θ = 15°

2θ = 30°

cos 30° = square root of 32

1 − cos 30°

1 − square root of 32

22 − square root of 32

2 − square root of 32

1 + cos 30°

1 + square root of 32

22 + square root of 32

2 + square root of 32

Divide.

2 − square root of 322 + square root of 32

2 − square root of 32 · 22 + square root of 3

2 − square root of 32 + square root of 3

Multiply the numerator and the denominator by 2 − square root of 3.

(2 − square root of 3)(2 − square root of 3)

4 − 4square root of 3 + (square root of 3)²

(square root of 3)² = 3

4 − 4square root of 3 + 3

7 − 4square root of 3

(2 + square root of 3)(2 − square root of 3)

4 − 3

The denominator is 1.

7 − 4square root of 3

Answer: 7 − 4square root of 3

1
Find the exact value of tan² 22.5°.
Show Answer

tan² θ = 1 − cos 2θ1 + cos 2θ

θ = 22.5°

2θ = 45°

cos 45° = square root of 22

1 − cos 45°

1 − square root of 22

22 − square root of 22

2 − square root of 22

1 + cos 45°

1 + square root of 22

22 + square root of 22

2 + square root of 22

Divide.

2 − square root of 222 + square root of 22

2 − square root of 22 · 22 + square root of 2

2 − square root of 22 + square root of 2

Multiply the numerator and the denominator by 2 − square root of 2.

(2 − square root of 2)(2 − square root of 2)

4 − 4square root of 2 + (square root of 2)²

(square root of 2)² = 2

4 − 4square root of 2 + 2

6 − 4square root of 2

(2 + square root of 2)(2 − square root of 2)

4 − 2

6 − 4square root of 22

Divide the numerator and the denominator by 2.

3 − 2square root of 2

Answer: 3 − 2square root of 2

2
Find the exact value of tan² 75°.
Show Answer

tan² θ = 1 − cos 2θ1 + cos 2θ

θ = 75°

2θ = 150°

cos 150° = −square root of 32

1 − cos 150°

1 − (−square root of 32)

1 + square root of 32

22 + square root of 32

2 + square root of 32

1 + cos 150°

1 + (−square root of 32)

1 − square root of 32

22 − square root of 32

2 − square root of 32

Divide.

2 + square root of 322 − square root of 32

2 + square root of 32 · 22 − square root of 3

2 + square root of 32 − square root of 3

Multiply the numerator and the denominator by 2 + square root of 3.

(2 + square root of 3)(2 + square root of 3)

4 + 4square root of 3 + (square root of 3)²

(square root of 3)² = 3

4 + 4square root of 3 + 3

7 + 4square root of 3

(2 + square root of 3)(2 − square root of 3)

4 − 3

The denominator is 1.

7 + 4square root of 3

Answer: 7 + 4square root of 3

3
Find the exact value of sin⁴ 15°.
Show Answer

sin⁴ θ = (sin² θ)²

sin² θ = 1 − cos 2θ2

θ = 15°

2θ = 30°

cos 30° = square root of 32

1 − cos 30°

1 − square root of 32

22 − square root of 32

2 − square root of 32

Divide by 2.

2 − square root of 322

2 − square root of 32 · 12

2 − square root of 34

Square that value.

(2 − square root of 3)(2 − square root of 3)

4 − 4square root of 3 + (square root of 3)²

(square root of 3)² = 3

4 − 4square root of 3 + 3

7 − 4square root of 3

7 − 4square root of 316

Answer: 7 − 4square root of 316

4
Find the exact value of cos⁴ 22.5°.
Show Answer

cos⁴ θ = (cos² θ)²

cos² θ = 1 + cos 2θ2

θ = 22.5°

2θ = 45°

cos 45° = square root of 22

1 + cos 45°

1 + square root of 22

22 + square root of 22

2 + square root of 22

Divide by 2.

2 + square root of 222

2 + square root of 22 · 12

2 + square root of 24

Square that value.

(2 + square root of 2)(2 + square root of 2)

4 + 4square root of 2 + (square root of 2)²

(square root of 2)² = 2

4 + 4square root of 2 + 2

6 + 4square root of 2

6 + 4square root of 216

Divide the numerator and the denominator by 2.

3 + 2square root of 28

Answer: 3 + 2square root of 28

5
cos 2θ = −35. Find sin² θ.
Show Answer

sin² θ = 1 − cos 2θ2

cos 2θ = −35

1 − cos 2θ

1 − (−35)

1 + 35

55 + 35

85

Divide by 2.

852

85 · 12

810

45

Answer: 45

6
cos 2θ = −513. Find cos² θ.
Show Answer

cos² θ = 1 + cos 2θ2

cos 2θ = −513

1 + cos 2θ

1 + (−513)

1 − 513

1313 − 513

813

Divide by 2.

8132

813 · 12

826

413

Answer: 413

7
cos 2θ = 1213. Find tan² θ.
Show Answer

tan² θ = 1 − cos 2θ1 + cos 2θ

cos 2θ = 1213

1 − cos 2θ

1 − 1213

1313 − 1213

113

1 + cos 2θ

1 + 1213

1313 + 1213

2513

1132513

113 · 1325

125

Answer: 125

8
cos 2θ = 2029. Find tan² θ.
Show Answer

tan² θ = 1 − cos 2θ1 + cos 2θ

cos 2θ = 2029

1 − cos 2θ

1 − 2029

2929 − 2029

929

1 + cos 2θ

1 + 2029

2929 + 2029

4929

9294929

929 · 2949

949

Answer: 949

9
Find the exact value of sin² 15° cos² 15°.
Show Answer

sin² θ = 1 − cos 2θ2

cos² θ = 1 + cos 2θ2

Multiply those two formulas.

1 − cos 2θ2 · 1 + cos 2θ2

(1 − cos 2θ)(1 + cos 2θ)

1 − cos² 2θ

1 − cos² 2θ4

θ = 15°

2θ = 30°

cos 30° = square root of 32

cos² 30° = (square root of 32)²

(square root of 3)²2²

(square root of 3)² = 3

2² = 4

34

1 − cos² 30°

1 − 34

44 − 34

14

Divide by 4.

144

14 · 14

116

Answer: 116

10
cos 2θ = −725. Find sin² θ.
Show Answer

sin² θ = 1 − cos 2θ2

cos 2θ = −725

1 − cos 2θ

1 − (−725)

1 + 725

2525 + 725

3225

Divide by 2.

32252

3225 · 12

3250

1625

Answer: 1625

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