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Chapter 6: Problem 76

Describe two strategies that can be used to verify identities.

Short Answer

Expert verified
Two strategies for verifying identities include algebraic manipulation where you manipulate one side of the identity to match the other, and substitution where you substitute specific values into the identity and confirm if both sides are equal. Additionally, graphical analysis, where two identity's graphs are compared, can also be effective especially for more complex identities.

Step by step solution

01

Strategy 1: Algebraic Manipulation

One common method to verify identities is by algebraic manipulation. In this strategy, start with one side of the identity and use algebraic rules to manipulate the equation to resemble the other side. This includes methods like combining like terms, factoring, applying distributive laws, and utilizing trigonometric identities, among other things. The purpose is to simplify or rewrite the equation such that it matches the other side, thus proving that they are indeed identical.
02

Strategy 2: Substitution

Another strategy that can be used to verify identities is substitution. This strategy involves substituting specific values into the variables of the identities and then checking if both sides of the identity are equal. While this method can be quick and effective, it's important to note that it doesn't necessarily prove the identity for all possible values. Therefore, it's often used as a secondary check or when dealing with specific values of variables.
03

Strategy 3: Graphical Analysis

A third strategy to verify identities, though not as commonly used as the first two, is through graphical analysis. When two mathematical expressions are equal, their graphs will be identical. Therefore, graph both sides of the equation on the same graph. If they look the same or overlap completely, it verifies the identity. Of course, this method requires that you can accurately graph the functions, which might be more complex for certain mathematical identities.

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Most popular questions from this chapter

Exercises \(110-112\) will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 110 to answer each of the following. a. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} ?\) b. Is \(\sin \left(2 \cdot 30^{\circ}\right),\) or \(\sin 60^{\circ},\) equal to \(2 \sin 30^{\circ} \cos 30^{\circ} ?\)

solve each equation on the interval \([0,2 \pi) .\) $$ |\sin x|=\frac{1}{2} $$

In the interval \([0,2 \pi),\) the solutions of \(\sin x=\cos 2 x\) are \(\frac{\pi}{6}, \frac{5 \pi}{6},\) and \(\frac{3 \pi}{2},\) Explain how to use graphs generated by a graphing utility to check these solutions.

Find the exact value of each expression. Do not use a calculator. $$ \cos \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(-\frac{1}{2}\right)\right] $$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$ 2 \sin 3 x+\sqrt{3}=0 $$
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