Question

Find sin (A + B) and sin (A - B) if sin A = 15/17 and cos B = 9/15

Answer 1

`\sf \sin(A + B) = (77)/(85)`

`\sf \sin(A - B) = (13)/(85)`

Explanation:

To find
`\sf \sin(A + B)` and
`\sf \sin(A - B)`, we can use the sum and difference formulas for sine:

`\sf \sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B`

`\sf \sin(A - B) = \sin A \cdot \cos B - \cos A \cdot \sin B`

Given that
`\sf \sin A = (15)/(17)` and
`\sf \cos B = (9)/(15)`, we need to find
`\sf \cos A` and
`\sf \sin B`.

First, let's find
`\sf \cos A` using the fact that
`\sf \sin^2 A + \cos^2 A = 1`:

`\sf \cos A = √(1 - \sin^2 A)\\ = \sqrt{1 - \left((15)/(17)\right)^2} \\= \sqrt{(17^2 - 15^2)/(17^2)}\\ = \sqrt{(289 - 225)/(289)}\\ = \sqrt{(64)/(289)}\\ = (8)/(17)`

Now, let's find
`\sf \sin B` using the fact that
`\sf \sin^2 B + \cos^2 B = 1`:

`\sf \sin B = √(1 - \cos^2 B) \\= \sqrt{1 - \left((9)/(15)\right)^2}\\ = \sqrt{1 - (81)/(225)} \\= \sqrt{(225 - 81)/(225)} \\= \sqrt{(144)/(225)} \\= (12)/(15)`

Now, we can substitute these values into the sum and difference formulas:

`\sf \sin(A + B) = (15)/(17) \cdot (9)/(15) + (8)/(17) \cdot (12)/(15)`

`\sf \sin(A - B) = (15)/(17) \cdot (9)/(15) - (8)/(17) \cdot (12)/(15)`

Let's simplify these expressions:

`\sf \sin(A + B) = (135 + 96)/(255) = (231)/(255) = (77)/(85)`

`\sf \sin(A - B) = (135 - 96)/(255) = (39)/(255) = (13)/(85)`