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Chapter 5: Problem 69

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot ^{2} x-9=0$$

Short Answer

Expert verified
The solutions are \(x \approx 0.3218, 2.8198, 3.4634\), and \(5.9614\).

Step by step solution

01

Rewrite The Equation

Start by rewriting the equation to isolate the cotangent squared value like this: \(\cot ^{2} x = 9\)
02

Calculate The Cotangent of x

The next step is to find the value of \(\cot x\). To do this, take the square root on both sides. Remember that when you take the square root of any number, the result can be both positive and negative. Hence, \(\cot x = \pm \sqrt {9}\), so \(\cot x = \pm 3\)
03

Use The Inverse Cotangent Function

In order to find the values of \(x\), we use inverse cotangent function on both sides of the equation. \(\cot ^{-1}(\cot x) = \cot ^{-1}(\pm 3)\), this simplifies to \(x = \cot ^{-1}(3)\) and \(x = \cot ^{-1}(-3)\). Remember that cotangent is the reciprocal of tangent and both functions have their period as \(\pi\). Hence, for any \(k\) in the set of integers, \(k \in \mathbb{Z}\), \(x = \cot ^{-1}(3) + k\pi\) and \(x = \cot ^{-1}(-3) + k\pi\)
04

Calculation And Checking The Interval

Calculate \(x = \cot^{-1}(3)\) and \(x = \cot^{-1}(-3)\) which might be outside the defined interval [0,2蟺). For each of these values of \(x\), you need to add multiples of \(\pi\) and check, if they're within the required interval. As it happens \(x = \cot^{-1}(3) \approx 0.3218\) and \(x = \cot^{-1}(-3) \approx \pi - 0.3218 \approx 2.8198\). \(x\) can also be \(x = \cot^{-1}(3) + \pi \approx 3.4634\) and \(x = \cot^{-1}(-3) + 2\pi \approx 5.9614\).
05

Listing All The Solutions

Gather the solutions which lie within the interval [0,2蟺). Therefore the solutions of the equation \(\cot ^{2} x -9 = 0\) in the interval [0,2蟺) are \(x \approx 0.3218, 2.8198, 3.4634\), and \(5.9614\).

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