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

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.