# Angle Sum Formulas for Sine and Cosine

- Author:
- Dr. Doug Davis, 3D

- Topic:
- Angles, Cosine, Sine, Trigonometry

The angle sum formulas for sine and cosine are developed here. Use the advance button at the bottom to show the steps in the development.
and change colors if they are negative.

- Two adjustable sliders allow you to change the angles in the ranges
and - Draw a right triangle with
at the origin - Draw another right triangle along the hypotenuse of the first triangle so the angles add
- Set a scale to the triangles by setting the hypotenuse of the second triangle to 1
- The adjacent side is then
- The opposite side is then
- Similarly for the first triangle, the adjacent side is then
times the hypotenuse - The opposite side is then
times the hypotenuse - Draw a third right triangle by extending the opposite side of the first triangle
- Note that the first angle is
- Calculate the adjacent side length
- Calculate the opposite side length
- Add another right triangle to complete the rectangle
- Note the angle that is
because of the opposite angles between parallel lines - The adjacent side is on top
- The opposite side is on the side

With the side lengths shown it can be seen that
and
.
Note that can be negative resulting in the angle difference formulas and . Include the relations from odd and even symmetry respectively.