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Chapter 5: Problem 12

Solve the equation. $$\tan x+\sqrt{3}=0$$

Short Answer

Expert verified
The solution to the given equation \(\tan x + \sqrt{3} = 0\) is \(x = k\pi + \frac{4\pi}{3}\) and \(x = k\pi + \frac{7\pi}{3}\) where \(k\) is an integer.

Step by step solution

01

Rearrange the equation

Rearrange the equation to isolate \(\tan x\) on one side by subtracting \(\sqrt{3}\) from both sides of the equation. The equation becomes \(\tan x = -\sqrt{3}\).
02

Solve for x

Knowing the basic values of the tan function, we can deduce that \(\tan x = - \sqrt{3}\) when \(x = \frac{4\pi}{3} \) or \(x = \frac{7\pi}{3}\) within one complete circle (inside the range from 0 to \(2\pi\)). However, since the \(\tan\) function has a period of \(\pi\), these solutions will repeat every \(\pi\).
03

Write the final solution

The solutions to the equation in the interval of one complete circle are \(x = \frac{4\pi}{3}, \frac{7\pi}{3}\). For a complete set of solutions, we have to add all multiples of \(\pi\) which gives the final result as \(x = k\pi + \frac{4\pi}{3}\) and \(x = k\pi + \frac{7\pi}{3}\) where \(k\) is an integer.

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