Use the angle difference identity to find the exact value of each. 3) cos 75 0 4) sin 105 0 5) show sin 4θ = 4sin θ cos θ cos 2 θ 6) solve 5 cos 2 θ = 1 for 0 o < θ < 360 o 7) solve cos 2 θ = cos θ for 0 o < θ < 360 o . 8) solve cos 2 θ = cos θ + 2 for 0 < θ < 2p

3) cos 75 = (√6 - √4)/4  

4) sin 105 = (√6 + √4)/4  

5) below

6) θ = 39.2° or 140.7°

7) θ = 120° or 240°

8) θ = 0.267π rad or π rad

Step-by-step explanation:

3) cos 75 = cos (30+45) = cos30.cos45 - sin30sin45=

√3/2 . √2/2 - 1/2 . √2/2 = (√6 - √4)/4

4)sin 105 = sin (60+45) = sin60.cos45 + sin45.cos60 =

√3/2 . √2/2 + 1/2 . √2/2 = (√6 + √4)/4

5)sin4θ = 4sinθcosθcos2θ

sin4θ = sin(2θ+2θ) = sin2θcos2θ + sin2θcos2θ = 2sin2θcos2θ

As sin2θ = sinθcosθ + sinθcosθ = 2sinθcosθ

substituting:

2.2sinθcosθ.cos2θ = 4sinθcosθcos2θ

6) 5cos2θ = 1               0°<θ<360°

cos2θ = 1/5

As cos2θ = cosθcosθ - sinθsinθ = cos²θ - sin²θ

and sin²θ + cos²θ = 1 → sin²θ = 1- cos²θ

we can say that cos2θ = cos²θ - (1- cos²θ) = cos²θ - 1 + cos²θ = 2cos²θ -1

So, 2cos²θ - 1 = 1/5

2cos²θ = 1/5 + 1

2cos²θ = 6/5

cos²θ = 6/10

cos²θ = 3/5

cosθ = ± √(3/5)

θ = +cos⁻¹√(3/5)  → θ = 39.2°

θ = -cos⁻¹√(3/5)  → θ = 140.7°

7) cos2θ = cosθ                    0°<θ<360°

From the item above, we know that cos2θ = 2cos²θ - 1

2cos²θ - 1 = cosθ

2cos²θ - cosθ - 1 = 0

Making cosθ = y to facilitate:

2y² - y - 1 = 0

Δ = (-1)² - 4.2.(-1) = 9

√Δ = 3

y = (1±3)/4

y₁ = 4/4 = 1

y₂ = -2/4 = -1/2

cosθ = y

cosθ = 1 → θ = 0°

cosθ = -1/2 → θ = 120° or 240°

As  0°<θ<360° (no equal sign) → θ = 120° or 240°

8) cos2θ = cosθ + 2 for 0 < θ < 2p

From the item above, we know that cos2θ = 2cos²θ - 1

2cos²θ - 1 = cosθ + 2

2cos²θ - cosθ - 3 = 0

Making cosθ = y to facilitate:

2y² - y - 3 = 0

Δ = (-1)² - 4.2.(-3) = 25

√Δ = 5

y = (1±5)/4

y₁ = 6/4 = 2/3

y₂ = -4/4 = -1

cosθ = y

cosθ = 2/3 → θ = 48.2° = 0.267π rad

cosθ = -1 → θ = 180° = π rad

$6.42

step-by-step explanation: