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How to Calculate the Sine of Special Angles in Radians

You can use the concept of sum and difference formulas to calculate the sine of special angles in radians. This process is different than solving equations because here you're asked to find the trig value of a specific angle that isn't readily marked on the unit circle

image0.png

Prior to choosing the appropriate formula, you simply break the angle into either the sum or the difference of two angles from the unit circle. Refer to the unit circle and notice the angles in radians in the figure.

The unit circle showing angles in radians with common denominators
The unit circle showing angles in radians with common denominators

You see that all the denominators are different, which makes adding and subtracting them a nightmare. You must find a common denominator so that adding and subtracting is a dream. The common denominator is 12, as you can see in the figure.

This figure comes in handy only for sum and difference formulas, because finding a common denominator is something you do only when you're adding or subtracting fractions.

As an example, follow these steps to find the exact value of sin(pi/12):

  1. Rewrite the angle in question, using the special angles in radians with common denominators.

    From the figure, you want a way to add or subtract two angles so that, in the end, you get pi/12.

    image2.png
  2. Choose the appropriate sum or difference formula:

    image3.png

    Because you rewrote the angle with subtraction, you need to use the difference formula.

  3. Plug the information you know into the chosen formula.

    You know the following equality:

    image4.png

    Substitute as follows into the difference formula:

    image5.png

    Which gives you

    image6.png
  4. Reduce the fractions in the formula to ones you're more comfortable with.

    In this example, you can reduce to

    image7.png

    Now you'll have an easier time referring to the unit circle to get your equation.

  5. Use the unit circle to look up the sine and cosine values that you need.

    You now have

    image8.png
  6. Multiply and simplify to get your final answer.

    You end up with the following answer:

    image9.png
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