Question

For positive acute angles A and B, it is known that sin A = 8/17 and cos B = 24/25

Find the value of sin(A + B) in simplest form.

Answer 1

Explanation:

Here we use the SUM FORMULA: sin(a + b) = sin a cos b + cos a sin b

sin(A + B) = sin A*cos B + cos A*sin B. We are not given cos A or sin B and so must find them using the Pythagorean Theorem x^2 + y^2 = z^2.

Looking at sin A = 8/17, we see that the "opposite side" is 8 and the "hypotenuse" is 17. Then (adjacent side) = √(17² - 8²) = 15; that is, the side adjacent to Angle A is 15. Thus, cos A = 15/17 and sin A = 8/17.

We find sin B similarly. cos B = (side adjacent to B)/25 = 24/25. Therefore the side opposite B is √(25² - 24²) = 7. Thus, sin B = 7/25 and cos B = 24/25.

Then sin (A + B) = sin A*cos B + cos A*sin B

= (8/17)*(24/25) + (15/17)(7/25)

Here there are two products of fractions. This product (above) can be rewritten as

8(24) + 15(7) (3)(64) + (3)(5)(7)

= sin A*cos B + cos A*sin B = --------------------------- = --------------------------

17(25) 17(25)