Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The solutions for the equation are \(x = \cot^{-1}(\sqrt{1/3}) + k\pi\) , where \(k = 0, 1\). Note that the solution is valid within the interval \([0, 2\pi]\).

Step 1: Rearrange the Equation

Begin the process by rearranging the equation in order to isolate the cotangent squared term. Add 1 to both sides of the equation to get: \(3 \cot^2 x = 1\), Then isolate \(\cot^2 x\), by dividing both sides by 3. Thus: \(\cot^2 x = 1/3\)

Step 2: Root to the Equation and Inverse Cotangent

Next, to remove the square, take the square root of both sides: \(\cot x = \sqrt{1/3}\). To solve for \(x\), take the cotangent inverse or arccotangent of both sides: \(x = \cot^{-1}(\sqrt{1/3})\)

Step 3: Finalize the Solution

Lastly, the final step is to calculate the cotangent inverse of \(\sqrt{1/3} = \cot^{-1}(\sqrt{1/3})\) to yield the answer. Remember that there are two possible solutions for \(x\) within the interval \([0, 2\pi]\) since cotangent function repeats every \(\pi\). Therefore obtain both solutions by adding \(\pi\) to the first solution: \(x = \cot^{-1}(\sqrt{1/3}) + k\pi\), for \(k = 0, 1\).