Chapter Four Overview

 The sine definition basically says that, on a right triangle, the following measurements are related the measurement of one of the non-right angles, the length of the side opposite to that angle and the length of the triangle's hypotenuse. Alternately, the cosine definition basically says that, on a right triangle, the following measurements are related: the measurement of one of the non-right angles, the length of the side adjacent to that angle and the length of the triangle's hypotenuse. Therefore cosine equals adjacent over hypotenuse and sine equals opposite over hypotenuse. In chapter four we examine conditional trigonometric equations, that is, equations that are true for only certain values of the variable. When solving for the solutions for these equations there might be an infinite amount of answers because of the periodicity of the trigonometric functions. Also, to verify a trigonometric identity we use the fundamental trigonetric identities, the even-odd properties, and the basic arithmetic and algebraic operations. In order to verify a trigonometric identity, we are required to show that the given expressions are equivalent. 

Trigonometric Equations

In earlier sections we examined trigonometric identities, which are equations involving trigonometric functions that are satisfied by all values of the variable for which both sides of the equality are defined. In this section we examine conditional trigonometric equations, that is, equations that are true for only certain values of the variable. When solving for the solutions for these equations there might be an infinite amount of answers because of the periodicity of the trigonometric functions. In general, to obtain solutions of an equation such as sin x=radical 2/2, it is more convenient to use a unit circle and reference angles rather than a graph of the trigonometric function. When solving an equation, if you divide by an expression containing a variable, you may lose some solutions of the original equation. To prevent the loss of a solution you must determine the values that make the expression zero and check to see whether they are solutions of the original equation. 

Verifying Identities

A trigonometric identity is an equation or formula involving only trigonometric functions that is valid for all angles measured in degrees or radians or for real numbers for which both sides of the equality are defined. To verify a trigonometric identity we use the fundamental trigonetric identities, the even-odd properties, and the basic arithmetic and algebraic operations. In order to verify a trigonometric identity, we are required to show that the given expressions are equivalent. Some suggestions for verifying identities includes: simplifying the more complicated side of the equation first and finding least common denominators for sums or differences of fractions. Also if the two preceding suggestions fail, then express all trigonometric functions in terms of sines and cosines and try to simplify. In the end, don't treat a trigonometric equation as an identity until after you have proven that it is really true.