Basic Pythagorean Identities for Trigonometry Functions
How to Find a Common Denominator of a Fraction to Solve a Trig Identity
Reciprocal Trigonometry Identities

Dealing with Half-Angle Identities Involving Radicals

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:

  1. Determine which angle is double the angle you’re working with.

    image0.png
  2. Substitute the angle measure into one of the half-angle tangent identities.

    image1.png
  3. Fill in the function values and simplify the answer.

    image2.png
  4. To get the radical out of the denominator, rationalize it by multiplying both parts of the fraction by the radical.

    image3.png

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

image4.png

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.

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How to Use the Double-Angle Identity for Sine
Change to Sines and Cosines in a Trigonometry Identity
The Origin of the Half-Angle Identities for Sine
How to Square Both Sides to Solve a Trigonometry Identity Problem
Product-to-Sum Identities
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