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Chapter 5: Problem 22

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

Short Answer

Expert verified
The solutions to the equation are \(x=\frac{\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{4\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{2\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{5\pi}{9}+\frac{k\pi}{3}\) where k is an integer.

Step by step solution

01

Isolate the Tangent Squared Function

To begin the process of isolating the variable 'x', first take the square root of both sides of the equation. This gives \(\tan3x = \sqrt{3}\) and \(\tan3x = -\sqrt{3}\). The negative solution is due to the fact that \(\sqrt{x^2}=x\) and \(-x\).
02

Apply Tangent Function Properties

The next step is to apply the properties of the tangent function. From our knowledge of trigonometric functions, we know that \(\tan(\frac{\pi}{3})=\sqrt{3}\) and \(\tan(\frac{4\pi}{3})=\sqrt{3}\). Also, \(\tan(\frac{2\pi}{3})=-\sqrt{3}\) and \(\tan(\frac{5\pi}{3})=-\sqrt{3}\). Hence the equations to solve now are \(3x=\frac{\pi}{3}+k\pi\), \(3x=\frac{4\pi}{3}+k\pi\), \(3x=\frac{2\pi}{3}+k\pi\) and \(3x=\frac{5\pi}{3}+k\pi\) where k is an integer.
03

Solve for x

Finally, divide every equation by 3 to find the values of x. This gives: \(x=\frac{\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{4\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{2\pi}{9}+\frac{k\pi}{3}\), \(x=\frac{5\pi}{9}+\frac{k\pi}{3}\).

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