cos (θ + x) is 16/65
Step-by-step explanation:
Given cos θ = -4/5, find sin θ.
cos θ = adjacent side/hypotenuse = -4/5
The opposite side can be found using Pythagoras Theorem.
Hypotenuse² = Opposite Side² + Adjacent Side²
⇒ Opposite side² = Hypotenuse² - Adjacent Side²
= 5² - (-4)² = 25 - 16 = 9
∴ Opposite Side = 3
⇒ sin θ = opposite side/hypotenuse = 3/5
Given sin x = -12/13, find cos x.
sin x = opposite side/hypotenuse = -12/13
The adjacent side can be found using Pythagoras Theorem.
Hypotenuse² = Opposite Side² + Adjacent Side²
⇒ Adjacent side² = Hypotenuse² - Adjacent Side²
= 13² - (-12)² = 169 - 144 = 25
∴ Adjacent Side = 5
⇒ cos x = adjacent side/hypotenuse = 5/13
Find cos (θ + x).
cos (θ + x) = cos θ cos x - sin θ sin x
= -4/5 × 5/13 - 3/5 × -12/13
= -20/65 + 36/65 = 16/65
