Find the exact value of the following under the given conditions a.cos(a-b) b.(a-b) c.tan(a-b)

Question

cos b=-1/2, pi<β<3pi/2

sin a=-5/7, pi/2<β<pi

Answer 1

Some known trig identities:

sin (a + b) = sin a cos b + cos a sin b, and sin (a - b) = sin a cos b - cos a sin b

cos (a + b) = cos a cos b - sin a sin b, and cos (a - b) = cos a cos b + sin a sin b

It's not stated what the Greek letter beta represents; we'll assume it refers to b in the first equation and a in the second. Also, I'm assuming that question b. is asking for the sin (a - b), not just (a - b). For that, you could simply express using inverse functions (arc cos and arc sin).

cos b = -1/2, in Quadrant III, so sin b = -(sqrt 3)/2

sin a = -5/7, in Quadrant II, so cos a = -(sqrt 24)/7 = -2(sqrt 6)/7

Now plug in the values into the identity formulas:

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

= (sqrt 6)/7 + (5/14) sqrt 3

= (2 sqrt 6 + 5 sqrt 3)/14

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

= 5/14 - (sqrt 6 sqrt 3)/7

= 5/14 - 3 (sqrt 2)/7

= (5 - 6 sqrt 2)/14

c. tan (a - b) = (5 - 6 sqrt 2)/(2 sqrt 6 + 5 sqrt 3)

This is very tedious, and it's easy to make mistakes with signs, sq rts, etc.