X8577 Find the exact value of the expression 3 8 17 7 cos 1

X)8.5.77 Find the exact value of the expression 3 8 17 7 -+ cos 1 sin sin 25 8 17 17 sin sin+ cos (Simplify your answer, including any radicals, Use integers or fra

Solution

Solution:

The identity for sin (a + b) = sin(a) cos(b) + sin(b) cos(a)

with a = sin^-1 (7/25) and b = cos^-1 (-8/17),

making sin(a) = sin(sin^-1 (7/25)) = 7/25 ==> sin(a) = 7/25

and cos(b) = cos(cos^-1 (-8/17)) = -8/17 ==> cos(b) = -8/17

By identity sin^2 (a) + cos^2 (a) = 1 you can get that

=> (7/25)^2 + cos^2 (a) = 1 => cos^2 (a) = 576/625 => cos(a) = 24/25

=> sin^2 (b) + (-8/17)^2 = 1 => sin^2 (b) = 225/289 => sin(b) = 15/17

Thus,

sin[sin^-1(7/25) + cos^-1(-8/17)] = sin[a + b] = sin(a) cos(b) + sin(b) cos(a)

                                              = (7/25) (-8/17) + (15/17) (24/25)

                                              = 304/425