Question

If cos A = 1/2, then what is the positive value of cos 1/2 A, in simplest radical form with a rational denominator?

Answer 1

`\cos((1)/(2)A) = {(√(3))/(2)`

Explanation:

Given

`\cos A = (1)/(2)`

Required

Determine
`\cos((1)/(2)A)`

To do this, we make use of the following identity

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

Substitute:
`\cos A = (1)/(2)`

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

Solve the numerator

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

`\cos((1)/(2)A) = \sqrt{((3)/(2))/(2)}`

Rewrite as:

`\cos((1)/(2)A) = \sqrt{(3)/(2) * (1)/(2)}`

`\cos((1)/(2)A) = \sqrt{(3)/(4)}`

Take square roots

`\cos((1)/(2)A) = {(√(3))/(2)`