91Ó°ÊÓ

Write down the given equation: \(\tan 2x = -1\).

To solve the equation \(\tan 2x = -1\), we need to find the general solution for the tangent function: \(2x = n\pi + (-1)^n \cdot \frac{\pi}{4}\), where n is an integer.

Now we need to divide the equation by 2 to solve for x: \(x = \frac{n\pi}{2} + (-1)^n \cdot \frac{\pi}{8}\).

We are interested in finding the values of \(x\) that satisfy the equation in the given range: \(0^{\circ} \leq x < 360^{\circ}\) or \(0 \leq x < 2\pi\). To do this, we need to substitute different integer values for n and check if the resulting x values fall within the given range: For \(n=0\), we have \(x = \frac{\pi}{8}\) or \(22.5^{\circ}\). For \(n=1\), we have \(x = \frac{\pi}{2} - \frac{\pi}{8} = \frac{3\pi}{8}\) or \(67.5^{\circ}\). For \(n=2\), we have \(x = \pi + \frac{\pi}{8} = \frac{9\pi}{8}\) or \(202.5^{\circ}\). For \(n=3\), we have \(x = \frac{3\pi}{2} - \frac{\pi}{8} = \frac{11\pi}{8}\) or \(247.5^{\circ}\). For \(n=4\), the value of \(x\) will be greater than \(2\pi\) (ie, \(x\) will fall outside the given range). Therefore, there are four possible solutions in the given range: \(x = 22.5^{\circ}\), \(67.5^{\circ}\), \(202.5^{\circ}\), and \(247.5^{\circ}\).