Question

How do you use the half-angle identity to find the exact value of cos (11pi/8)?

Answer 1

Find `cos ((11pi)/8)`

Ans:`-sqrt(2 - sqrt2)/2`

Explanation:

`cos ((11pi)/8) = cos ((3pi)/8 + pi) = -cos (3pi)/8`= cos t
Use trig identity: `cos 2t = 2cos^2 t - 1`
`cos 2t = cos ((6pi)/8) = cos ((3pi)/4) = -sqrt2/2`
`-sqrt2/2 = 2cos^2 t - 1`
`2cos^2 t = 1 -sqrt2/2 = (2 - sqrt2)/2`
`cos^2 t = (2 - sqrt2)/4`
`cos t = cos ((11pi)/8) = +- sqrt(2 - sqrt2)/2.`
Only the negative answer is accepted because the arc (11pi/8 is in Quadrant II.
`cos ((11pi)/8) = - sqrt(2 - sqrt2)/2`
Check by calculator.
`cos ((11pi)/8) = cos 247.5` deg = - 0.382
`- sqrt(2 - sqrt2)/2 =` - 0.382. Correct.