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How do you use the half-angle identity to find the exact value of cos(-pi/12)?

Answer 1

#sqrt(2+sqrt3)/2#

#cos(-pi/12)=cos(pi/12)##as# #cos(x)# #is# #a# #even function# # f(x)=f(-x)# #costheta=sqrt((1+cos2theta)/2)# #=>cos(pi/12)=sqrt((1+cos(pi/6))/2)=sqrt((1+sqrt3/2)/2)=sqrt(2+sqrt3)/2#

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Answer 2

Daniel Acosta

To find the exact value of (\cos\left(-\frac{\pi}{12}\right)) using the half-angle identity, follow these steps:

Start with the half-angle identity for cosine: (\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1+\cos(\theta)}{2}}).
Substitute (\theta = -\frac{\pi}{6}) into the identity.
Calculate (\cos\left(-\frac{\pi}{6}\right)) using known values (or the unit circle).
Plug the value of (\cos\left(-\frac{\pi}{6}\right)) into the half-angle identity.
Simplify the expression to find the exact value of (\cos\left(-\frac{\pi}{12}\right)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.