Derive Sum of Two Angles Identities (Unit Circle)

This example derives the sum of two angle identities using the circle definitions of the trigonometric functions sine and cosine on the unit circle.

Steps

  1. Start by drawing the point corresponding to sum of two angles (alpha) and (beta) on the unit circle. From the circle definitions of the trigonometric functions we know the horizontal coordinate of this point is equal to and the vertical coordinate is equal to .

    Sum of two angles identities

    This gives us the initial setup to derive the identities, where the goal is to express and in terms of the trigonometry of the individual angles and .

  2. Next, draw the trigonometry related to the angles and as two right triangles stacked on top of eachother. I’ll also draw the length on top of the figure to make room for other expressions.

    Sum of two angles identities

    The key insight here is to draw a vertical line through the right-corner vertex of the right-triangle corresponding to the angle . This forms two right-triangles that are similar to the right-triangle formed by the angle . The vertical and horizontal component of the point formed by then can be represented using the side-lengths of these two right-triangles.

    Sum of two angles identities
  3. From the unit circle we know that the adjacent side of the right triangle formed by is equal to and the opposite side is equal to .

    Sum of two angles identities

    Since we now know the hypotenuse of this first similar right triangle, we can solve for the adjacent and opposite sides using the circle definitions of the trig functions. Note, you can also imagine scaling the adjacent side and opposite side by the value .

    Sum of two angles identities

    Repeat the same process to find the lengths of the second similar right triangle. This time the side lengths are scaled by the value .

    Sum of two angles identities
  4. Finally, equate the vertical and horizontal lengths together to derive the identities.

    Sum of two angles identities

    This results in the sum of two angles identities shown below.