SOLUTION: If cos&#952; = sqrt((1/2) + (1/2 sqrt 2)) and sin&#952; = sqrt((1/2) - (1/2 sqrt 2)) with 0<=&#952;<2&#960;, it follows that 2&#952; = ________&#960;

Algebra ->  Trigonometry-basics -> SOLUTION: If cos&#952; = sqrt((1/2) + (1/2 sqrt 2)) and sin&#952; = sqrt((1/2) - (1/2 sqrt 2)) with 0<=&#952;<2&#960;, it follows that 2&#952; = ________&#960;       Log On


   

Question 1037478: If cosθ = sqrt((1/2) + (1/2 sqrt 2)) and sinθ = sqrt((1/2) - (1/2 sqrt 2)) with 0<=θ<2π, it follows that 2θ = ________π

Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
If cosθ = sqrt((1/2) + (1/2 sqrt 2)) and sinθ = sqrt((1/2) - (1/2 sqrt 2)) with 0<=θ<2π, it follows that 2θ = ________π
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

We are given 

cos%28theta%29 = sqrt%281%2F2+%2B+1%2F%282%2Asqrt%282%29%29%29,  sin%28theta%29 = sqrt%281%2F2+-+1%2F%282%2Asqrt%282%29%29%29,  and  0 <= theta < 2pi.

Notice that the given data implies that theta is in Q1, since both  cos%28theta%29  and  sin%28theta%29  are positive.


Let us calculate   sin%282theta%29.  It is

sin%282theta%29 = 2%2Asin%28theta%29%2Acos%28theta%29 =  =  = 

2%2Asqrt%28%281%2F2%29%5E2+-+%281%2F%282%2Asqrt%282%29%29%29%5E2%29 = 2%2Asqrt%281%2F4-1%2F8%29 = 2%2Asqrt%281%2F8%29 = 1%2Fsqrt%282%29 = sqrt%282%29%2F2.    (1)


Similarly,  cos%282theta%29 = 2%2Acos%5E2%28theta%29-1    (well known formula for those who might be interested in such a problem)  ---> 

= 2%2A%28sqrt%281%2F2+%2B+1%2F%282%2Asqrt%282%29%29%29%29%5E2-1 = 2%2A%281%2F2%2B1%2F%282%2Asqrt%282%29%29%29-1 = 1%2Fsqrt%282%29 = sqrt%282%29%2F2.    (2)

It follows from (1) and (2) that 2theta%29 = pi%2F4.

Hence, theta = pi%2F8.
Solved.