Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula.

Question

a) square root 1-cos(80°)/2

b)square root 1-cos(6𝜃)/2

I am having a hard time finding the exact answer to these. Steps and an answer would be highly appreciated.

Thank you

Answer 1

sin(θ/2) = ±√[(1 - cosθ) / 2], where the + or - is determined by the quadrant in which θ/2 lies.

So, sin40° = sin(80°/2) = √[(1 - cos80°)/2] and sin(3θ) = sin(6θ/2) = √[(1 - cos(6θ))/2], provided that 3θ lies in either quadrant 1 or quadrant 2.

Answer 2

a) (1-cos(80°)/2)^.5=

(1-.1736/2)^.5=

(.8263)^.5=.909038.........

^ is the power sign.