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.
- 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.