Sum of two Angles Identities

The sum of two angles are two trigonometric identities that express the cosine and sine of the sum of two angles in terms of their individual components. The two identities are given in the equations below:

Visual Proof

Sum of Two Angles Identites

The two identities can be visualized as a “proof-without-words” shown above.

Derive Sum of Two Angles Identities
Sum of Two Angles Identity Goal

To derive the sum of two angles identities the length of and the length of can be calculated using the properties of a right triangle with a hypotenuse of length 1. Two right triangles can be chosen for and so that the lengths can be expressed in terms of their sides.

Steps
  1. Start by placing two right-triangles of angles and next to eachother so that their angles sum together.

    Sum of Two Angles Identity Goal

    There are a couple of choices of how to relate these two right-triangles together. Note in this choice, the right triangle defined by is scaled by . Ordinary with a hypotenuse of length the length of the adjacent side would be and the length of the opposite side would be , but since the hypotenuse is , they are both scaled by the value of .

  2. Then, we can redraw the triangle defined by the angle and label the sides for reference and so they are out of the way.

    Sum of Two Angles Identity Goal

    Observe that if we find the two unknown lengths denoted by question marks in the image below, we have all the information needed to express the length of and .

    Sum of Two Angles Identity with unknown lengths labeled

    These two lengths are given by the equations:

  3. Observe that the right-triangle whose lengths we are interested in also has an angle of . Then, we can find the length of each of its sides since we know the length of its hypotenuse which is .

    Sum of Two Angles Last Triangle

    This gives us the figure below with all the lengths labeled:

    Sum of Two Angles with Everything Labeled

    Finally, we can substitue these found lengths into the equations from step 2 to get the two identities: