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How to Find Trigonometric Functions of an Angle by Using Pythagorean Identities

You can use Pythagorean identities to find the trig function of an angle if you know one trig function of the angle and are looking for another. For example, if you know the sine of an angle, you can use the first Pythagorean identity to find the cosine of the angle. In fact, you can find whatever you're asked to find if all you have is the value of one trig function and the understanding of what quadrant the angle θ is in.

The three Pythagorean identities are

image0.png

Here's an example:

If you know that

image1.png

by following these steps:

  1. Plug what you know into the appropriate Pythagorean identity.

    Because you're using sine and cosine, you use the first identity:

    image2.png

    Plug in the values you know to get

    image3.png
  2. Isolate the trig function with the variable on one side.

    First square the sine value to get 576/625, giving you

    image4.png

    Subtract 576/625 from both sides (Hint: You need to find a common denominator):

    image5.png
  3. Figure the square root both sides (taking both the positive and negative square roots) to solve.

    You now have

    image6.png

    But you can have only one solution because of the constraint

    image7.png

    you're given in the problem.

  4. Draw a picture of the unit circle so you can visualize the angle.

    Because

    image8.png

    the angle lies in quadrant II, so the cosine of θ must be negative. You have your answer:

    image9.png
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