How do you evaluate cos((13pi)/8)?

2 Answers
P dilip_k
May 14, 2016

=1/2sqrt((2-sqrt2)

Explanation:

using formula costheta=sqrt(1/2(1+cos(2theta)))
cos((13pi)/8)
=sqrt(1/2(1+cos(2*(13pi)/8)))

=sqrt(1/2(1+cos((13pi)/4)))

=sqrt(1/2(1+cos(3pi+pi/4))

=sqrt(1/2(1-cos(pi/4)))

=sqrt(1/2(1-1/sqrt2))

=sqrt((sqrt2-1)/(2sqrt2)

=sqrt((2-sqrt2)/(2xx2)

=1/2sqrt((2-sqrt2)

Nghi N. ยท Nghi N
May 14, 2016

sqrt(2 - sqrt2)/2

Explanation:

cos ((13pi)/8) = cos ((3pi)/8 + cos ((16pi)/8)) = cos ((3pi)/8 + 2pi) =
= cos ((3pi)/8).
Evaluate cos ((3pi)/8) by applying the trig identity:
cos 2a = 2cos^2 a - 1.
In this case, we get -->
2cos^2 ((3pi)/8) - 1 = cos ((6pi)/8) = cos ((3pi)/4) = -sqrt2/2
2cos^2 ((3pi)/8) = 1 - sqrt2/2 = (2 - sqrt2)/2
cos^2 ((3pi)/8) = (2 - sqrt2)/4
cos ((3pi)/8) = +- sqrt(2 - sqrt2)/2
cos ((13pi)/8) = cos ((3pi)/8) = sqrt(2 - sqrt2)/2 , because
cos ((3pi)/8) is positive.

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