How do you use the half angle identity to find exact value of cos(2pi/5)?

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Answer 1 · 2015-09-27T19:20:25

Refer to explanation

Notice that #sin(3*pi/5)) = sin(2*pi/5))#

The triple angle sine formula is:# sin(3x) = 4 sin(x) cos^2(x) - sin(x)#

The double angle sine formula is: #sin(2x) = 2 sin(x) cos(x)#

Since these are equal when #x = pi/5#

#4 sin(x) cos^2(x) - sin(x) = 2 sin(x) cos(x)#

Divide out the sin(x) term:

#4 cos^2(x) - 1 = 2 cos(x)#

This gives a simple quadratic in #cos(x)#, substituting
#y = cos(pi/5)#:

#4y^2 - 2y - 1 = 0 #

Solving for y gives 2 solutions: #y = (1 + sqrt(5))/4# and #y = (1 - sqrt(5))/4#. Since cos(pi/5) is positive: #cos(pi/5) = (1 + sqrt(5))/4#

Use the double angle formula: #cos(2x) = 2 cos^2(x) - 1#

#cos(2pi/5) = 2*(1 + sqrt(5)/4)^2 - 1 = (sqrt(5) - 1)/4#