How to Use Half-Angle Identities to Find the Sine of an Angle

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.

    image0.png

    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.

    image1.png

    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.

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