Verified answer

tan A = 8/15

opp =8

adj = 15

hyp = sqrt(8^2+15^2) = sqrt(64+225) = sqrt(289) = 17 (Pythogorean theorem)

sin A = opp/hyp = -8/17

cos A = adj /hyp = -15/17

(negative because sin and cos are negative in quadrant III π<A<3π/2)

cos B = adj/hyp

adj = 12

hyp = 13

opp = sqrt(13^2-12^2) = sqrt(169-144) = sqrt(25) = 5 (Pythogoren theorem)

sin B = opp/hyp = -5/13

(negative because sin is negative and cos is positive in quadrant IV 3π/2<B<2π)

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

sin(A+B) = (-8/17)(12/13) + (-15/17)(-5/13)

=-21 /221

One point on A can be represented by (-15,-8), or the complex number α = -15-8i, where |α| = 17.

Similarly, one point on B can be represented by (12,-5), or β = 12-5i, where |β| = 13.

So αβ = (-180-40) + (75-96)i = -220-21i, and |αβ| = |α||β| = 17*13 = 221

Sin(A+B) = Im(αβ)/|αβ| = -21/221 is the correct answer.

sin A cos B + cos A sin B

( - 8/17) x 12/13 + (-15/17) x ( - 5/13) = - 21/ 221

Ans: You are incorrect

Hint:

Tan A = 8/15 , π<A<3π/2 --->sinA = -8/17 , cosA = -15/17

CosB = 12/13, 3π/2<B<2π --->sinB=-5/13

sin(a)/cos(a) = 8/15

15 * sin(a) = 8 * cos(a)

225 * sin(a)^2 = 64 * cos(a)^2

225 - 225 * cos(a)^2 = 64 * cos(a)^2

225 = 289 * cos(a)^2

+/- 15 = +/- 17 * cos(a)

+/- 15/17 = cos(a)

Since we're in Q3, then cos(a) = -15/17

sin(a) = (8/15) * cos(a) = (8/15) * (-15/17) = -8/17

cos(b) = 12/13

sin(b)^2 + cos(b)^2 = 1

sin(b)^2 + 144/169 = 1

sin(b)^2 = 25/169

sin(b) = +/- 5/13

b is in Q4, so sin(b) = -5/13

sin(a) = -8/17

cos(a) = -15/17

sin(b) = -5/13

cos(b) = 12/13

sin(a + b) = sin(a)cos(b) + sin(b)cos(a)

(-8/17) * (12/13) + (-15/17) * (-5/13) =>

(15 * 5 - 8 * 12) / (17 * 13) =>

(75 - 96) / 221 =>

-21/221