Derive Difference of Two Angles Identities

This example shows how to derive the difference of two angles identities using the right triangle definitions of the functions sine and cosine. The definitions are shown below.

Steps

  1. Start by drawing a right triangle with an angle of and a hypotenuse of . The angles (alpha) and (beta) are also drawn.

    derive difference of two angles identities step 1 a

    Solve for the lengths of the adjacent and opposite side. Substitute the hypotenuse , adjacent side and opposite side into the defintions of sine and cosine.

    Label these lengths in the figure.

    derive difference of two angles identities step 1 b

    The goal is to represent these side-lengths in terms of the sine and cosine of the angles and . To achieve this goal, draw another right triangle of hypotenuse with the angle of on top of the first right-triangle.

    derive difference of two angles identities step 1 b

    This allows us to represent the length of as the sum of the lengths and . And to represent the length as the difference of the lengths and .

    Substituting the expressions from above gives us the starting point for the identities. In the next steps we will solve for the unknown lengths using the definitions of sine and cosine.

  2. Solve for the adjacent and opposite sides of the right-triangle illustrated below.

    derive difference of two angles identitites step 2 a

    Substitute the hypotenuse , adjacent side and opposite side into the definitions of sine and cosine, then solve for the adjacent and opposite side.

    Label the side lengths in the figure.

    derive difference of two angles identitites step 2 b

  3. Solve for the adjacent and opposite sides of the right-triangle illustrated below. From the corresponding angle theorem we know that is the same as .

    derive difference of two angles identitites step 3 a

    Substitute the hypotenuse , adjacent side and opposite side into the definitions of sine and cosine, then solve for the adjacent and opposite side.

    Label these lengths in the figure.

    derive difference of two angles identitites step 3 b

  4. Solve for the adjacent and opposite sides of the right-triangle illustrated below. We know that is equal , because it is complementary to which is complementary angle to .

    derive difference of two angles identitites step 4 a

    Substitute the hypotenuse , adjacent side and opposite side into the defintions of sine and cosine and solve for the adjacent and opposite side.

    Label these lengths into the illustration below.

    derive difference of two angles identitites step 4 a

    This gives us all the unknown lengths in the figure.

    derive difference of two angles identitites step 4 c

  5. Substitute the lengths into the equation from the end of step 1.

    This give us the difference of two angles identities.

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