Question

Hi! Can some please explain how to evaluate these two questions, so that I can understand how to complete them?

1. Given: cos∅= -4/5, sin x= -12/13, ∅ is in the third quadrant, and x is in the fourth quadrant; evaluate cos(theta - x).
2. Given: sin∅= -3/5, ∅ is a third quadrant angle, and tanФ= -7/24, Ф is a second-quadrant angle; find tan(∅+Ф).

Thank you for your help!

Answer 1

Answer:

1. 16/65

2. 44/117

Step-by-step explanation:

1. cos(θ−x)

Start with angle difference formula.

cos θ cos x + sin θ sin x

-4/5 cos x − 12/13 sin θ

Use Pythagorean identity to find cos x and sin θ.

sin² x + cos² x = 1

(-12/13)² + cos² x = 1

cos² x = 25/169

cos x = ±5/13

Since x is in the fourth quadrant, cos x > 0.  So cos x = 5/13.

sin² θ + cos² θ = 1

sin² θ + (-4/5)² = 1

sin² θ = 9/25

sin  θ = ±3/5

Since θ is in the third quadrant, sin θ < 0.  So sin θ = -3/5.

Plug values in and solve:

-4/5 (5/13) − 12/13 (-3/5)

-20/65 + 36/65

16/65

2. tan(θ+φ)

Use angle sum formula:

(tan θ + tan φ) / (1 − tan θ tan φ)

(tan θ − 7/24) / (1 + 7/24 tan θ)

Use Pythagorean theorem to find cos θ.

sin² θ + cos² θ = 1

(-3/5)² + cos² θ = 1

cos² θ = 16/25

cos θ = ±4/5

Since θ is in the third quadrant, cos θ < 0.  So cos θ = -4/5.

Therefore, tan θ = sin θ / cos θ = (-3/5) / (-4/5) = 3/4.

(3/4 − 7/24) / (1 + (7/24) (3/4))

(18/24 − 7/24) / (1 + 7/32)

(11/24) / (39/32)

(11/24) (32/39)

44/117