By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions. For example, you can use the half-angle identity when the exact value of the trig function involves radicals.
This example uses tan π/8:
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Determine which angle is double the angle you’re working with.
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Substitute the angle measure into one of the half-angle tangent identities.
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Fill in the function values and simplify the answer.
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To get the radical out of the denominator, rationalize it by multiplying both parts of the fraction by the radical.
The other identity for the tangent of a half angle gives you exactly the same answer. That form isn’t any easier, though, because both the sine and cosine of this angle have a radical in them. If the problem involved an angle of 60 degrees, though, the story would be different. The sine of 60 degrees is
and the cosine is 1/2 , which practically begs you to use the form with the cosine in the denominator so you don’t have to mess with a radical in the denominator. Both identities work — the one you use is just a matter of personal preference.