Find the exact values of sin(a + b) and cos(a − b).

Question

Given that sin(a) = 3/4 and cos(b) = -1/5 , with a and b both in the interval [π/2 , π)

Answer 1

Using the Angle Addition/Subtraction Identities:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b) so to calculate this we need these 4 items:

sin(a), cos(b), cos(a), sin(b)

We are given sin(a) and cos(b) so we need to calculate cos(a) and sin(b). We are also given that angle a is in Q2

Since sin(a) = opp/hyp then the opposite side is 3 and the hypotenuse is 4 like this:

Using the Pythagorean Theorem, we can calculate "x":

x = √(4² - 3²) = √(16 - 9) = √7 BUT, since this is in Q2 the √7 MUST be negative (going left of the origin)

So cos(a) = -√7/4

Since cos(b) = adj/hyp = -1/5, then the adjacent side = -1 and the hypotenuse = 5 like this:

Again, using the Pythagorean Theorem, y = √(5² - 1²) = √(25 - 1) = √24 = 2√6

So sin(b) = 2√6/5

So

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

sin(a + b) = (3/4)(-1/5) + (-√7/4)(2√6/5)

sin(a + b) = -3/20 + -2√42/20

sin(a + b) = -(3 + 2√42)/20

For the second question, we use: cos(a – b) = cos(a)cos(b) + sin(a)sin(b) and plug in the vales from above.