Given : cos (θ) = - 4/5 | sin x = - 12/13

Evaluate: cos (θ + x)

cos (θ + x) is 16/65

Step-by-step explanation:

Step 1:

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

Step 2:

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

Step 3:

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