Question

Find the exact value by using a
half-angle formula.
cos 75° =
2-√3

Answer 1

`\frac{\sqrt{2-√(3)}}{2}`

Explanation:

since 75 is half of 150, we can use the half angle formula of cos defined as:
`cos((\theta)/(2)) = \sqrt{(1+cos(\theta))/(2)}\\` where theta=150

So plugging in the values we get:

`cos(75) = \sqrt{(1+cos(150))/(2)}`

Now using the unit circle, we can solve for cos(150)

`cos(75) = \sqrt{(1-(√(3))/(2))/(2)\\`

Now multiply the 1 by 2/2 to combine the numerator into one fraction

`cos(75) = \sqrt{((2)/(2)-(√(3))/(2))/(2)\\`

Combine the numerator into one fraction

`cos(75) = \sqrt{((2-√(3))/(2))/(2)\\`

Keep, change, flip

`cos(75) = \sqrt{(2-√(3))/(2)*(1)/(2)}`

Multiply:

`cos(75) = \sqrt{(2-√(3))/(4)`

We can distribute the square root across the division:

`\frac{\sqrt{2-√(3)}}{√(4)}`

Simplify the denominator

`\frac{\sqrt{2-√(3)}}{2}`