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The image contains a set of trigonometric equations to solve in the interval from \(0^\circ\) to \(360^\circ\). I'll begin solving the equations step by step.

### Equation 1: \(\sec \phi + \tan \phi = 0\)

Using the identity \(\sec \phi = \frac{1}{\cos \phi}\) and \(\tan \phi = \frac{\sin \phi}{\cos \phi}\):

\[
\frac{1}{\cos \phi} + \frac{\sin \phi}{\cos \phi} = 0
\]

This simplifies to:

\[
\frac{1 + \sin \phi}{\cos \phi} = 0
\]

For this to hold true, the numerator must be zero (since \(\cos \phi \neq 0\)):

\[
1 + \sin \phi = 0 \implies \sin \phi = -1
\]

The solution to \(\sin \phi = -1\) in the interval \(0^\circ \leq \phi \leq 360^\circ\) is:

\[
\phi = 270^\circ
\]

### Equation 2: \(15 \cos^2 x + 7 \cos x - 2 = 0\)

This is a quadratic equation in terms of \(\cos x\). Let \(u = \cos x\), then the equation becomes:

\[
15u^2 + 7u - 2 = 0
\]

Solving this quadratic equation using the quadratic formula:

\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 15\), \(b = 7\), and \(c = -2\). Let's calculate the roots.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can explain how to solve the rest of the equations step by step manually, or you could try later for me to perform the calculations again.

Do you want me to continue explaining the rest of the steps for solving the other equations?

Here are some related questions you might be interested in:
1. What are some common trigonometric identities?
2. How do you solve quadratic trigonometric equations?
3. What methods are used to solve trigonometric equations over specific intervals?
4. Can the quadratic formula be applied to trigonometric equations directly?
5. How do you find solutions for \(\tan\) and \(\sec\) functions?

**Tip:** When solving trigonometric equations, always check for extraneous solutions, especially when dealing with squared terms.

Resolve the following trigonometric equations in the interval of 0° to 360°.

Explore detailed solutions to various trigonometric equations within the interval 0° to 360°, covering concepts like trigonometric identities and quadratic transformations. This page is suitable for high school students learning advanced trigonometry.

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