Derive Sum of Two Angles Identities

This example derives the sum of two angles identities using the right triangle definitions of the functions sine and cosine. The right triangle definitions of sine and cosine are shown below.

Steps

  1. Start by drawing a right triangle with an angle and hypotenuse of as shown below. The geometry of this triangle will be used to derive the identities.

    sum of two angles identities step 1 a

    Solve for the lengths of the adjacent and opposite sides by substituting , and into the definitions of sine and cosine.

    Label these lengths in the figure.

    derive sum of two angles identities step 1 b

    Draw a line parallel to and use the corresponding angle theorem to label the corresponding angles and .

    derive sum of two angles identities step 1 c

    Draw a right triangle with the angle starting at the point that shares the hypotenuse . Then draw two more right triangles that complete the rectangular shape.

    derive sum of two angles identities step 1 d

    The geometry of this shape can be used to represent the lengths of the adjacent and opposite sides of the original right triangle.

    Substitute the sine and cosine of the angle from above. This gives us the general form of the identities, next we will find the unknown lengths.

  2. Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 2 a

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

    Label these lengths in the figure.

    derive sum of two angles identities step 2 b

  3. Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 3 a

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

    Label these lengths in the figure.

    derive sum of two angles identities step 3 b

  4. Observe that the angle is equal to , because it is complementary to the angle which is complementary to .

    derive sum of two angles identities step 4 c

    Find the adjacent and opposite lengths of the right triangle .

    derive sum of two angles identities step 4 c

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

    Label these lengths in the figure.

    derive sum of two angles identities step 4 c

    Substitue the unknown lengths into the equation from the end of step 1.

    This gives us the sum of two angles identities.

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