Question

Find the exact value by using a half-angle identity.cos five pi divided by twelve one divided by two times the square root of the quantity two plus the square root of three one divided by two times the square root of the quantity two minus the square root of three negative one divided by two times the square root of the quantity two plus the square root of three negative one divided by two times the square root of the quantity two minus the square root of three

Answer 1

+√[(2 - √3)/4]

Explanation:

Here is the complete question

How do you find the exact values of cos(5pi/12) using the half angle formula?

Solution

The half-angle formula is

cos(θ/2) = ±√[(1 + cosθ)/2]

if cos(5π/12) = cos(θ/2) ⇒ θ/2 = 5π/12 ⇒ θ = 5π/6

So, θ = 5π/6 × 180/π = 150°

Cos150 = cos(180 - 30) = -cos30 = -cosπ/6 = -√3/2

So, cos(5π/12) = ±√[(1 + cos5π/6)/2]

= ±√[(1 + (-cosπ/6))/2]

= ±√[(1 - cosπ/6)/2]

= ±√[(1 - √3/2)/2]

= ±√[(2 - √3)/2 ÷ 2]

= ±√[(2 - √3)/4]

Since 5π/12 = 5π/12 × 180/π = 75° which is in the first quadrant, so

cos(5π/12) = +√[(2 - √3)/4]. We ignore the negative answer