By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions using the functions of angles you already know. For example, even though you can use a difference identity to find the sine of 15 degrees by finding the sine of the difference between 45 and 30 degrees, you can also use the half-angle identity.

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

    Twice 15 is 30, so the choice is 30 degrees. Stick to the more-common angles — the ones that have exact values or are multiples of 30 and 45.

  2. Substitute that angle into the half-angle identity for sine.

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    Because the sine of 15 degrees is a positive value, the sign in front of the radical becomes +.

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

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    The result isn’t a particularly pretty value, although beauty is in the eye of the beholder. Some would consider this answer to be wonderful, because it’s the exact value and not a decimal approximation.