The development of all the trigonometric identities relies on the use of other identities and the special relationship with the unit circle. This discovery is noticeable in the proofs for the trigonometric identities mentioned above. To find the sum and difference formulas of sine and cosine the Pythagorean identities had to be proved and used in the proof. This is the same for the development of the double angle formulas. To prove the double angle formulas the sum and difference formulas were used and manipulated through algebra.
Trigonometry has a special relationship with the unit circle and right triangles. Moore and LaForest mentioned the importance of this relationship in understanding trigonometry and the development of their identities with this relationship. “If students are to use trigonometric functions productively, they must understand angle measure, the unit circle, and right triangles in ways that let them see trigonometric functions as relations between two quantities.” Points on the unit circle are described by the angle that is formed by the line segment between the center and the point and the x-axis. As the angle increases or decreases the point on the circle changes because it is traveling around the circle. The use of triangles are used to describe show the changes occurring at these points (Moore & LaForest, 2014).
This is done by inscribing right triangles into the unit circle, where the vertex of the angle in the triangle is constructed as the center of the unit circle. This inscription changes the meaning of the hypotenuse to be a radius of the unit circle and the opposite leg is now a chord of the unit circle. Since the unit circle represents all circles then the right triangle inscribed in a unit circle is also dynamic. Therefore, all trigonometric functions represent the lengths of the sides of a the triangles with respect to the angle. This relationship is important for the basic proofs of the Pythagorean trigonometric identities and the sum and difference identities (Moore & LaForest, 2014) Use the applet below to explore this relationship of trigonometric functions and the unit circle.