Let's start by solving the two problems from the image you've provided:

Problem 1 (Solve for $-\pi \leq \theta \leq \pi$):

Equation:

$3 \tan \theta - 3 = 5 \tan \theta - 1$

Steps:

1. Move all terms involving $\tan \theta$ to one side: $3 \tan \theta - 5 \tan \theta = -1 + 3$ $-2 \tan \theta = 2$

2. Solve for $\tan \theta$: $\tan \theta = \frac{2}{-2} = -1$

3. Find $\theta$ such that $\tan \theta = -1$ in the interval $-\pi \leq \theta \leq \pi$:

   • The principal solution is $\theta = -\frac{\pi}{4}$, where $\tan \theta = -1$.
   • The general solution for $\tan \theta = -1$ is: $\theta = -\frac{\pi}{4} + n\pi \quad \text{for any integer } n.$ Considering the given interval $-\pi \leq \theta \leq \pi$, the solutions are: $\theta = -\frac{\pi}{4}, \quad \theta = \frac{3\pi}{4}.$

Thus, the solutions are: $\boxed{\theta = -\frac{\pi}{4}, \frac{3\pi}{4}}.$

Problem 2 (Solve for $-180^\circ \leq \theta \leq 90^\circ$):

Equation:

$2 \csc \theta = 6$

Steps:

1. Solve for $\csc \theta$: $\csc \theta = \frac{6}{2} = 3$

2. Recall that $\csc \theta = \frac{1}{\sin \theta}$, so: $\sin \theta = \frac{1}{3}$

3. Find $\theta$ such that $\sin \theta = \frac{1}{3}$ in the interval $-180^\circ \leq \theta \leq 90^\circ$.

   • The principal solution is $\theta = \arcsin\left(\frac{1}{3}\right) \approx 19.47^\circ$.
   • Since the sine function is positive in the first and second quadrants, the solution for the second quadrant within the given interval is $\theta = 180^\circ - 19.47^\circ = 160.53^\circ$, but this is outside the given range, so we discard it.

Therefore, the only solution is: $\boxed{\theta \approx 19.47^\circ}.$

Would you like further details on any of the steps? Here are 5 follow-up questions that relate to this content:

1. How can you derive the general solution for a trigonometric equation?
2. What is the significance of the reference angle when solving for trigonometric functions?
3. How does the periodicity of the tangent function affect the number of solutions in a given interval?
4. Can you explain how the sine function behaves in different quadrants?
5. How do you convert angles between radians and degrees?

Tip: When solving trigonometric equations, always consider the behavior of the function in each quadrant and any restrictions on the angle.