Google Classroom
GeoGebraGeoGebra Classroom

Angle Sum Formulas for Sine and Cosine

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.
  1. Two adjustable sliders allow you to change the angles in the ranges and
  2. Draw a right triangle with at the origin
  3. Draw another right triangle along the hypotenuse of the first triangle so the angles add
  4. Set a scale to the triangles by setting the hypotenuse of the second triangle to 1
  5. The adjacent side is then
  6. The opposite side is then
  7. Similarly for the first triangle, the adjacent side is then times the hypotenuse
  8. The opposite side is then times the hypotenuse
  9. Draw a third right triangle by extending the opposite side of the first triangle
  10. Note that the first angle is
  11. Calculate the adjacent side length
  12. Calculate the opposite side length
  13. Add another right triangle to complete the rectangle
  14. Note the angle that is because of the opposite angles between parallel lines
  15. The adjacent side is on top
  16. The opposite side is on the side
Note: The angles and change colors if they are negative.
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.