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Chapter 5: Problem 20

Solve the equation. $$\tan ^{2} 3 x=3$$

Short Answer

Expert verified
The solutions for the trigonometric equation \( \tan^{2} 3x = 3 \) are \( x = \frac{\pi}{9} + \frac{n\pi}{3} \) or \( x = -\frac{\pi}{9} + \frac{n\pi}{3} \) where n is an integer.

Step by step solution

01

Rewrite the Equation

Before proceeding, the equation needs to be rewritten in a more workable form. Rewrite the equation as \( \tan 3x = \sqrt{3} \) or \( \tan 3x = -\sqrt{3} \).
02

Solve for x using tan properties

Since \( \tan(\pi/3) = \sqrt{3} \) and \( \tan(-\pi/3) = -\sqrt{3} \), we can say 3x equals to \( \frac{\pi}{3} + n\pi \) or 3x equals to \( -\frac{\pi}{3} + n\pi \) where n is an integer.
03

Solve for x

Now, by dividing each equation by 3, obtain the solution of x. The solutions are: \( x = \frac{\pi}{9} + \frac{n\pi}{3} \) or \( x = -\frac{\pi}{9} + \frac{n\pi}{3} \) where n is an integer.

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