Question

If `α, β ∈ (0, π/2)`, sin α = `4/5` and cos (α + β) = `-12/13`, then sin β is equal to ______.

Options

• `63/65`

• `61/65`

• `3/5`

• `5/13`

Answer 1

If `α, β ∈ (0, π/2)`, sin α = `4/5` and cos (α + β) = `-12/13`, then sin β is equal to `underlinebb(63/65)`.

Explanation:

Given, cos (α + β) = `-12/13`

Here, 0 < (α + β) < π

∴ sin (α + β) = `sqrt(1 - cos^2(α + β)`

= `sqrt(1 - 144/169)`

= `5/13`

Now, sin β = sin [(α + β) – α]

= sin (α + β) cos α – cos (α + β) sin α

= `5/13 . 3/5 - (-12/13). 4/5`

= `15/65 + 48/65`

= `63/65`