Suppose the θ is an acute angle and cosθ = 4/5. Using the results sin2x = 1/2*(1 - cos2x) and cos2x = 1/2*(1 + cos2x), find the exact values of cos1/2θ, sin1

Question

Suppose the θ is an acute angle and cosθ = 4/5. 

 

Using the results sin²x = 1/2*(1 - cos²x) and cos²x = 1/2*(1 + cos2x),

 

find the exact values of cos1/2θ, sin1/2θ, and tan1/2θ 

Answer 1

The equations you need are sin²x = 1/2(1-cos2x) and cos²x = 1/2(1+cos2x)

 

Letting x = 1/2Θ, we get sin²(1/2θ) = 1/2(1-cosΘ) ... sin(1/2θ)=√((1-cosθ)/2) = √((1-4/5)/2) = √(1/10) = (√10)/10

 

To find the cos(1/2θ) use cos(1/2θ)=√((1+cosθ)/2)

 

I hope this helps.

Kevin