Technical Mathematics, Sixth Edition

Chapter 16. Trigonometric Identities and Equations

OBJECTIVES When you have completed this chapter, you should be able to Write a trigonometric expression in terms of the sine and cosine. Simplify a trigonometric expression using the fundamental identities. Prove trigonometric identities using the fundamental identities. Simplify expressions or prove identities using the sum or difference formulas, the double-angle formulas, or the half-angle formulas. Evaluate trigonometric expressions. Solve trigonometric equations.

In mathematics we usually try to simplify expressions as much as possible. In earlier chapters, we simplified algebraic expressions of all sorts. In this, our final chapter on trigonometry, we will simplify trigonometric expressions. For example, an expression such as tan x cos x simplifies to sin x.

For this we need to know how various trigonometric functions are related. We will start with the simplest (and most useful) fundamental identities. These identities are equations relating one trigonometric expression to another. Using them, we can replace one expression with another that will lead to a simpler result. We then proceed to trigonometric expressions containing sums and differences of two angles, double angles, and half angles.

This is followed by a short section on evaluating trigonometric expressions and another on solving trigonometric equations. We approximately found roots of a trigonometric equation by calculator in the preceding chapter, and here we learn ...