Sum and Difference Identities

The sum identities, in my opinion, are the first pillar identities in trigonometry. Most other identities can be derived from these two basic identities. These identities are also two of the least obvious, at least to me. "That the cosine or sine of a sum or difference is not the sum or difference of the cosines or sines is easy to establish by counterexample. However, the actual expansions for cos(A+B), cos(A-B), sin(A+B) and sin(A-B) are not intuitively obvious for most students. Standard textbooks do cover the derivations of these expansions, but the results have a greater impact if students conjecture them first." (Hurwitz, 2002, p. 510) I couldn't agree more. I think students need to see where these come from, and make their own conjectures.

While doing this project, I came across a geometric proof of the sum identities for sine and cosine. Below are applets demonstrating these proofs. I think there is power in seeing things geometrically. Guanshen Ren, a math teacher, said, "Studying trigonometric identities from a geometric perspective helps students make connections among various branches of mathematics and raise students' interests and appreciations in geometry." (1995, p.24) I have personally found these proofs very helpful. I think it important for students to understand that these came from somewhere specific, and to be able to use them. I am not a big fan of memorizing equations, but I do believe that the sum identities are two that would be worth committing to memory.

Geometric Proofs of the Sum Identities:

Cosine Sum Identity
Sine Sum Identity

Deriving the Tangent Sum Identity:

Difference Identities:



References: