Expert: Ahmed Salami - 1/27/2009

Question
If you could help me get started and run me through the steps of how to solve these it would be great. If you can reply before midnight PACIFIC TIME that would be great!
thanks.




Simplify the given expression. (Enter the exact answer as a fraction. Decimal answers will not be accepted. Your answer should not contain sin, cos, or tan.)



1. cos(pi/6 - x), if cos(x)= -1/6 and pi/2 < x < pi





2. cos(pi/3 + x), if sin(x)= -3/4 and 3pi/2 < x < 2pi







3. Simplify the given expression. (star (*) equals degrees)

sin(69*)sin(34*)-cos(69*)cos(34*)




I've tried these problems and I'm struggling to find the correct answer so i would be great to know what I'm doing wrong by seeing your process of solving. thanks.

Answer
Hi Brooke,
cos(A + B) = cosA.cosB - sinA.sinB
cos(A - B) = cosA.cosB + sinA.sinB
(cosx)^2 + (sinx)^2 = 1
cosx = sqrt[1 - (sinx)^2]
sinx = sqrt[1 - (cosx)^2]

1)cos(pi/6 - x) = cos(pi/6).cosx + sin(pi/6).sinx
cosx = -1/6
sinx = sqrt[1 - (-1/6)^2]
    = sqrt[1 - 1/36]
    = sqrt(35/36)
    = (sqrt35)/6
We take the positive square root because pi/2 < x < pi
cos pi/6 = (sqrt3)/2
sin pi/6 = 1/2
So,
cos(pi/6 - x) = (sqrt3)/2.(-1/6) + 1/2.(sqrt35)/6
             = -(sqrt3)/12 + (sqrt35)/12
             = (sqrt35 - sqrt3)/12

2)cos(pi/3 + x) = cos(pi/3).cosx - sin(pi/3).sinx
sinx = -3/4
cosx = sqrt[1 - (-3/4)^2]
    = sqrt[1 - 9/16]
    = sqrt(7/16)
    = (sqrt7)/4
We take the positive square root because 3pi/2 < x < 2pi
cos pi/3 = 1/2
sin pi/3 = (sqrt3)/2
So,
cos(pi/3 + x) = 1/2.(sqrt7)/4 - (sqrt3)/2.(-3/4)
             = (sqrt7)/8 + (sqrt3)/8
             = (sqrt7 + sqrt3)/8

3)sin(69)sin(34) - cos(69)cos(34)
= -[cos(69)cos(34) - sin(69)sin(34)]
= - cos(69 + 34)
= - cos(103)
= - [-cos(180 - 103)]
= cos 77

Regards