Answer:
1. 16/65
2. 44/117
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