Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation. $$\sqrt{3} \csc x-2=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(\sqrt{3}\csc x - 2 = 0\) are \(x=\frac{\pi}{3}\) and \(x=\frac{2\pi}{3}\).

Step 1: Isolate the Trigonometric Function

Rearrange the given equation \(\sqrt{3} \csc x - 2 = 0\) to isolate \(\csc x\) on one side. We get \(\csc x = \frac{2}{\sqrt{3}}\).

Step 2: Convert Cosecant to Sine

As cosecant is the reciprocal of sine, we can rewrite the equation in terms of sine. Hence, \(\sin x = \frac{\sqrt{3}}{2}\).

Step 3: Solve for x

The value of \(\sin x = \frac{\sqrt{3}}{2}\) happens at \(x = \frac{\pi}{3}\) and \(x = \frac{2\pi}{3}\) within the domain of \(0 \leq x < 2\pi \). So the solutions are \(x=\frac{\pi}{3}\) and \(x=\frac{2\pi}{3}\).