Calculus question

Question

I am having a really hard time figuring out this question. If anyone could help that would be much appreciated!!

Given that Cos A= -3/4, pi/2 < a < pi; Tan B= Sqrt 5/2, pi < B < 3pi/2

Find the exact value of Sin( a-b )

Answer 1

sin (a - b) = sin a cos b - cos a sin b, so we have to find the values of sin a, sin b, and cos b.

a is in the second quadrant, so sin a is positive

sin a = √(1 - cos² a) = √(1 - 9/16) = √7 / 4

b is in the third quadrant, so both sin b and cos b are negative.

1 + tan² b = sec² b, so sec² b = 9 / 4

cos² b = 4 / 9, so cos b = -2 / 3

sin b = cos b * tan b = -2 / 3 *√5 / 2 = - √5 / 3

sin (a - b) = sin a cos b - cos a sin b

= √7 / 4 * (-2 / 3) - ( -3 /4) * (- √5 / 3)

Answer 2

Since,

so

.

Since .

When , we have .

So

,

.

Then

.