Question

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

Answer 1

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