Evaluating using sum and difference identities

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

1) sin 165 = sin (120 + 45) = sin120 cos45 + cos120 sin 45 = (sqrt(3)/2)(-sqrt(2)/2)+ (-1/2)(sqrt(2)/2)

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

= -1/4 [ sqrt(6) + sqrt(2)]

2) 2 pi - 7/12 pi = pi*(24-7)/12 = (17/12)*pi

partial fraction decomposition

A/4 + B/3 = 17/12

3A + 4B = 17

A=3 B=2

9/12 pi = 3/4 pi and 2/3 pi makes 17/12 pi

cos ( 3/4 pi) cos (2/3 pi) - sin (3/4 pi) sin (2./3 pi) =

cos (135) cos ( 120) - sin (135) sin (120) =

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

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

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

(3)

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

cos(u) = -15/17 in quadrant 3 where it is negative

it is right triangle x = -8, y=-15, hypotenuse=17

sin(u) = -8/17, cos(u) = -15/17, tan(u) = 15/8

sin(v) = 4/5 in quadrant 1 where it is positive

it is triangle x=3,y=4,hypotenuse=5

sin(v) = 3/5, cos(v) = 4/5, tan(v) = 3/4

cos(u-v) = cos(u)cos(v) + sin(u)sin(v) = (-15/17)(4/5) + (-8/17)(3/5)

= -60/85 + -24/85

= -84/85