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Chapter 5: Problem 24

Solve the equation. $$\left(2 \sin ^{2} x-1\right)\left(\tan ^{2} x-3\right)=0$$

Short Answer

Expert verified
The solutions for the equation given by the exercise are \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{3}, -\frac{\pi}{3}\)

Step by step solution

01

Set the first factor equal to zero

Handle the first factor in the equation which is \(2 \sin ^{2} x-1=0\). To make it zero, solve for \(x\) by rearranging and simplifying: \(2 \sin ^{2} x = 1 \rightarrow \sin ^{2} x = 0.5 \rightarrow \sin x = \sqrt{0.5} \rightarrow x = \sin^{-1}(\sqrt{0.5})\). Solution set one using the domain of \(\sin x\) which is \(-\pi<=x<=\pi\) gives \(x_1 = \frac{\pi}{4}, x_2 = \frac{3\pi}{4}\)
02

Set the second factor equal to zero

The second part of the product is \(\tan ^{2} x - 3 = 0\). Setting this equal to zero and solving gives the equation: \(\tan ^{2} x = 3 \rightarrow \tan x = \sqrt{3} \). Solving for \(x\) gives: \( x= \tan^{-1}(\sqrt{3}) \). The domain for \(\tan x\) unlike \(\sin x\) is \(-\frac{\pi}{2}<x<\frac{\pi}{2}\). Thus, the solution set is \(x_3 = \frac{\pi}{3}\), \(x_4 = -\frac{\pi}{3}\)
03

Combine the results

After solving the two equations, we have four potential solutions for our original equation: \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{3}, -\frac{\pi}{3}\). As there is no overlap between these in the domains of \(\sin x\) and \(\tan x\) all of these solutions are valid.

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Most popular questions from this chapter

Verify the identity. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

Determine whether the statement is true or false. Justify your answer. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x+1}{1-\tan x}$$

Find the exact value of the expression. $$\frac{\tan 25^{\circ}+\tan 110^{\circ}}{1-\tan 25^{\circ} \tan 110^{\circ}}$4

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Write the trigonometric expression as an algebraic expression. $$\sin (\arcsin x+\arccos x)$$
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