Find cosa b if sina 817 where a is in the first quadrant a

Find cos(a + b) if sin(a) = 8/17 where a is in the first quadrant and tan(b) = -7/24 where b is in the second quadrant.

Solution

sin a = 8/17

cos a = +- ( 1- sin²a)^1/2

= + ( 1- (8/17)² )^1/2   ( +ve because cos is +Ve in I quadrant)

cos a = 0.88

tan b = -7/24

sec b = +/ ( 1 + tan²b)^1/2

= - ( 1+ ( -7/24)²)^1/2   ( -ve because sec is negative in II quadrant)

= - 1.04

cos b = 1/sec b

= - 1/1.04 = - 0.961

sin b = +- ( 1 - cos²b)^1/2

= + ( 1- ( -0.961)²)^1/2   ( + ve , because sin is +Ve in II quadrant)

= 0.276

cos ( a+b) = cos a* cos b - sina *sin b

= 0.88* ( - 0.961) - 0.470* ( 0.276)

= - 0.9754