Sharing Teaching Ideas: Cos_x + Sin_x and the Trigonometric Sum and Difference Identities

Abstract

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) 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. This article describes a lesson that introduces the formal proofs of the trigonometric sum and difference identities. In this activity, students discover the expansions for the cosine and sine of a sum and difference by using technology to generate selected graphs and by analyzing patterns in these graphs. The only prerequisite is the ability to find the equation of a sinusoid when the graph of the function is given.