Derive the Trigonometric Identities (Unit Circle)

This examples demonstrates how to derive the trigonometric identities using the trigonometric functions and the geometry of the unit circle. This examples contains how to:

  1. Derive Pythagorean Identity (Unit Circle)
  2. Derive Sum of Two Angles Identities (Unit Circle)
  3. Derive Difference of Two Angles Identities (Unit Circle)
  4. Derive Double Angle Identities (Unit Circle)
  5. Derive Half Angle Identities (Unit Circle)

Derive Pythagorean Identity (Unit Circle)

Pythagorean Identity

The pythagorean identity relates the squared sides of the right triangle on the unit circle together. However, it can be hard to see how the squared sides translates to the geometry of the right triangle on the unit circle. This example shows the steps to finding the relationship between the length of and the squared length of the and .

Steps

  1. Start with the right-triangle on the unit circle as shown below defined by the point . Label the lengths of the adjacent and opposite sides in terms of the angle of the triangle.

    derive pythagorean identity (unit circle) step 1

    The lengths of the adjacent and opposite side can be solved for by applying the definitions of the sine and cosine functions.

  2. Next, divide the right triangle into two similar triangles by drawing a line from the corner of the right-triangle perpendicular to its hypotenuse. This is shown below:

    derive pythagorean identity (unit circle) step 2 a

    Then find the length of the adjacent side, labeled with the variable , of the first similar triangle shown below:

    derive pythagorean identity (unit circle) step 2 b

    Apply the definition of cosine and then substitute the length of in for the hypotenuse and the length of for the adjacent side and then solve for .

    Repeat this process to find the length of the opposite side of the second similar triangle, labeled with the variable :

    derive pythagorean identity (unit circle) step 2 c

    Apply the definition of sine and then substitute the length of in for the hypotenuse and the length of for the opposite side and then solve for .

  3. Finally, we can observe that the hypotenuse the right triangle of lenght can be expressed as the sum of the lengths and which gives us pythagorean’s identity:

Derive Sum of Two Angles (Unit Circle)

This example shows how to derive the sum of two angle identities, shown below, using the geometry of the unit circle. This example is part of a series of examples the show how to derive the trigonometric identities using the unit circle.

Steps

  1. Start by drawing the cosine and sine lengths associated with the sum of two angles.

    A circle of radius one placed at the origin of the coordinate system.
    Figure 1: Unit Circle

    Test figure.

  2. Step number two.
  3. Step number three.

Derive Difference of Two Angles Identities (Unit Circle)

TODO:

Derive Double Angles Identities (Unit Circle)

TODO:

Derive Half Angles Identities (Unit Circle)

TODO:

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