Use sum and difference identities to evaluate trig functions

Question

Use sum and difference identities to evaluate trig functions

1) find the exact value of sin 165 degree

2) find the exact value of cos -7 pi / 12

3) find the exact value of cos ( u-v) given cos u = -15/17, where pi <u< 3pi/2, and sin v = 4/5, where 0<v<Pi/2

Answer 1

sin (165) = sin( 120 + 45) = sin(120) cos(45) + cos(120) sin(45)

= sqrt(3)/2 * sqrt(2)/2 + (-1/2) * sqrt(2)/2

= sqrt(6)/4 - sqrt(2)/2

= [sqrt(6) - 2*sqrt(2)] /4

cos ( - 7 pi/12) = cos ( (-1/2) pi) + (-1/12)pi)

= cos ( - 90 + -15) <--- degrees

= cos(-90) cos(-15) - sin(-90)sin(-15)

= cos(270) cos(315) - sin(270)sin(315)

= 0 * cos(315) - (-1)(sin 315)

= sin 315

= = sqrt(2)/2

cos u = -15/17 ---> sin u = -8/17

sin v = 4/5 ---> cos v = 3/5

cos(u-v) = cos(u) cos(v) + sin(u)sin(v)

= (-15/17)(3/5) + (-8/17)(4/5)

= -45/85 + -32/85

= -77/85