21. If sin θ = 1/4 and 0 ≤ θ ≤ π/2 ?

Question

(a) determine the EXACT value of cos 2θ

(b) determine the EXACT value of sin( θ/2)

Answer 1

a) cos2θ = 1 - 2sin²θ 

so, 1 - 2(1/4)² => 7/8

b) cosθ = 1 - 2sin²(θ/2)  

Now, cosθ = ±√15/4

As  0 ≤ θ ≤ π/2, cosθ = √15/4

so, √15/4 = 1 - 2sin²(θ/2)

i.e. 2sin²(θ/2) = (4 - √15)/4

=> sin²(θ/2) = (4 - √15)/8

Hence, sin(θ/2) = √[(4 - √15)/8] 

:)>

Answer 2

0 ≤ θ ≤ π/2 ---> cosθ >0 , sin( θ/2) >0

identity

(sinθ)^2 + ( cosθ)^2=1

and

formula

cos(2θ) = (cosθ)^2 - (sinθ)^2 = 1- 2(sin(θ))^2

you can do

Answer 3

Refer to the answers for the many similar questions you have recently posted.