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.[br][list=1][br][*] Two adjustable sliders allow you to change the angles in the ranges [math]0^\circ \le \alpha \le 180^\circ [/math] and [math]-180^\circ \le \beta \le 180^\circ [/math] [br][/*][*] Draw a right triangle with [math]\alpha[/math] at the origin[br][/*][*] Draw another right triangle along the hypotenuse of the first triangle so the angles add[br][/*][*] Set a scale to the triangles by setting the hypotenuse of the second triangle to 1[br][/*][*] The adjacent side is then [math]\cos \beta[/math][br][/*][*] The opposite side is then [math]\sin \beta[/math][br][/*][*] Similarly for the first triangle, the adjacent side is then [math]\cos \alpha[/math] times the hypotenuse[br][/*][*] The opposite side is then [math]\sin \alpha[/math] times the hypotenuse[br][/*][*] Draw a third right triangle by extending the opposite side of the first triangle[br][/*][*] Note that the first angle is [math]\alpha[/math][br][/*][*] Calculate the adjacent side length[br][/*][*] Calculate the opposite side length[br][/*][*] Add another right triangle to complete the rectangle[br][/*][*] Note the angle that is [math]\alpha + \beta[/math] because of the opposite angles between parallel lines[br][/*][*] The adjacent side is on top[br][/*][*] The opposite side is on the side[br][/*][/list][br]Note: The angles [math]\beta[/math] and [math]( \alpha +\beta )[/math]change colors if they are negative.

With the side lengths shown it can be seen that[br][math]\sin(\alpha + \beta) = \sin \alpha \:\cos \beta + \cos \alpha \:\sin \beta[/math][br]and[br][math]\cos( \alpha+\beta ) = \cos \alpha \:\cos \beta - \sin \alpha \:\sin \beta[/math].[br][br]Note that [math]\beta[/math] can be negative resulting in the angle difference formulas [math]\sin ( \alpha - \beta) [/math] and [math]\cos ( \alpha - \beta) [/math]. Include the relations [math]\sin\left(-\beta\right)=-\sin\left(\beta\right)\text{ and }\cos\left(-\beta\right)=\cos\left(\beta\right)[/math] from odd and even symmetry respectively.