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Chapter 5: Problem 16

Solve the equation. $$3 \cot ^{2} x-1=0$$

Short Answer

Expert verified
The solutions are \(x = \arctan \sqrt{3} + n\pi\) and \(x = \arctan -\sqrt{3} + n\pi\), where n is an integer.

Step by step solution

01

Rewrite equation

First, we need to rewrite the equation \(3 \cot ^{2} x-1=0\) in a workable format. Let's arrange the equation so it looks like this: \(3 \cot ^{2} x = 1\).
02

Isolate cotangent function

We then isolate the cotangent function. Divide both sides of the equation by 3: \(\cot ^{2} x = \frac{1}{3}\).
03

Square root both sides

Then, take the square root of both sides of the equation to obtain the cotangent of x. Remember that when we take the square root of any number, we get two results, positive and negative. So, we get: \(\cot x = 卤\sqrt{1/3}\).
04

Calculate the cotangent value

Find the value for \(\cot x\). Cotangent is the reciprocal of the tangent. So x is the angle whose tangent is the reciprocal of \(\sqrt{1/3}\), which gives two angles: \(x = \arctan \sqrt{3}\) and \(x = \arctan -\sqrt{3}\).
05

Find the general solution

Since cotangent function has a period of \(\pi\), the general solution will be \(x = \arctan \sqrt{3} + n\pi\) and \(x = \arctan -\sqrt{3} + n\pi\), where n is an integer

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