It seems you've uploaded an image with a mathematical problem. Let me process it for better clarity and provide a solution to the questions shown in the image.The image contains two problems. I'll analyze and solve them step by step.

Problem 1:

Find all degree solutions for the following: $\cos(5\theta) = 0$ Solution:

The cosine function equals zero at: $\theta = 90^\circ + 180^\circ k, \quad k \in \mathbb{Z}$

Here, $5\theta$ represents the argument: $5\theta = 90^\circ + 180^\circ k$

Divide by 5 to isolate $\theta$: $\theta = \frac{90^\circ}{5} + \frac{180^\circ k}{5}$

Simplify: $\theta = 18^\circ + 36^\circ k, \quad k \in \mathbb{Z}$

Since cosine is periodic, all solutions are represented by: $\theta = 18^\circ + 36^\circ k \quad \text{and} \quad \theta = 54^\circ + 36^\circ k$

Problem 2:

Find all solutions if $0 \leq x < 2\pi$: $\sec(3x) = -1$

Solution:

Recall that $\sec(x) = -1$ when $\cos(x) = -1$. Therefore, we solve: $\cos(3x) = -1$

The cosine function equals $-1$ at: $3x = \pi + 2n\pi, \quad n \in \mathbb{Z}$

Simplify for $x$ by dividing by 3: $x = \frac{\pi}{3} + \frac{2n\pi}{3}, \quad n \in \mathbb{Z}$

Find solutions in $0 \leq x < 2\pi$:

1. $x = \frac{\pi}{3}$
2. Add $2\pi/3$: $x = \frac{\pi}{3} + \frac{2\pi}{3} = \pi$
3. Add another $2\pi/3$: $x = \pi + \frac{2\pi}{3} = \frac{5\pi}{3}$

Thus, the solutions are: $x = \frac{\pi}{3}, \pi, \frac{5\pi}{3}$

Final Answers:

1. Problem 1: $\theta = 18^\circ + 36^\circ k \quad \text{and} \quad \theta = 54^\circ + 36^\circ k, \quad k \in \mathbb{Z}$
2. Problem 2: $x = \frac{\pi}{3}, \pi, \frac{5\pi}{3}$

Would you like further details or explanations?

Related Questions:

1. What are the general forms for sine and cosine equations for periodicity?
2. How does the secant function relate to the cosine function?
3. Can you calculate similar solutions for sine equations?
4. What happens to the range of solutions if the domain changes?
5. How do transformations like $5\theta$ or $3x$ affect trigonometric functions?

Tip: Always simplify trigonometric identities to connect them with standard results like zeros of cosine and sine.