How do you use the half-angle identity to find the exact value of cos [ - (3pi) / 8]?

Question

Zoe Austin · Accepted Answer

Find #cos ((-3pi)/8)#

Ans: #sqrt(2 - sqrt2)/2# Call cos ((-3pi)/8) = cos t #cos 2t = cos ((-6pi)/8) = cos ((6pi)/8) = cos ((3pi)/4) = -sqrt2/2# Use the trig identity: #cos 2t = 2cos^2 t - 1# #cos 2t = -sqrt2/2 = 2cos^2 t - 1# #2cos^2 t = 1 - sqrt2/2 = (2 - sqrt2)/2# #cos^2 t = (2 - sqrt2)/4# #cos t = cos ((-3pi)/8) = +- sqrt(2 - sqrt2)/2# Only the positive answer is accepted, because the arc #(-3pi)/8# is in Quadrant IV. Check by calculator. #sqrt(2 - sqrt2)/2# = 0.382 #cos ((3pi)/8) = cos 67.5# deg = 0.382.# Correct

Caroline Adams · Answer

undefined To use the half-angle identity to find the exact value of $ \cos\left(-\frac{3\pi}{8}\right) $:

1. Recognize that $ -\frac{3\pi}{8} $ is in the second quadrant.
2. Use the half-angle identity for cosine: $ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} $, where the sign depends on the quadrant.
3. Halve the angle: $ \frac{-3\pi}{8} $ becomes $ \frac{-3\pi}{16} $.
4. Since $ -\frac{3\pi}{8} $ is in the second quadrant, where cosine is negative, choose the negative sign.
5. Calculate $ \cos\left(\frac{-3\pi}{16}\right) = -\sqrt{\frac{1 + \cos\left(\frac{-3\pi}{8}\right)}{2}} $.
6. Substitute the value of $ \cos\left(\frac{-3\pi}{8}\right) $ using the half-angle identity.
7. Calculate $ \cos\left(\frac{-3\pi}{8}\right) = -\sqrt{\frac{1 + \cos\left(-\frac{3\pi}{4}\right)}{2}} $.
8. In the second quadrant, $ \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} $.
9. Substitute this value into the expression: $ \cos\left(\frac{-3\pi}{8}\right) = -\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} $.
10. Simplify to find the exact value.