Question

Cosθ = −√2 / 3, where π ≤ θ ≤ 3π / 2 .

tanβ = 4/3 , where 0 ≤ β ≤ π / 2 .
What is the exact value of sin(θ+β) ?
Enter your answer, as a fraction in simplified form.

sin(θ+β) =


I tried & I got -√7/5 - 4√2/15.

I'm pretty sure it's wrong, because it asks for a fraction in simplified form, but if someone could help that would be great.. thank you:)

Answer 1

(3√7-4√2)/15

Explanation:

Given Cosθ = −√2 / 3, where π≤θ≤3π/2 and tanβ = 4/3 , where

0≤β≤ π/2 .

In order to find sin(θ+β), we will expand it first

On expansion:

sin(θ+β) = sinθcosβ + cosθsinβ

If tanβ = 4/3

According to TOA:

Opposite = 4

Adjacent = 3

Hypotenuse = √4²+3²

Hypotenuse = √25

Hypotenuse = 5

Cosβ = Adjacent/Hypotenuse = 3/5

Sinβ = Opposite/Hypotenuse = 4/5

Similarly,

If cos θ = -√2/3

According to CAH

Adjacent = -√2

Hypotenuse = 3

To get the opposite:

3² = (-√2)²+opposite ²

9 = 2+ opposite²

Opposite² = 7

Opposite = √7

According to SOH,

Sinθ = opposite/hypotenuse = √7/3

Substituting all he given parameters into the expression:

sin(θ+β) = sinθcosβ + cosθsinβ

sin(θ+β) =√7/3(3/5) + (-√2/3)(4/5)

sin(θ+β) = 3√7/15 - 4√2/15

sin(θ+β) = √7/5 - 4√2/15

On simplification;

sin(θ+β) = (3√7-4√2)/15