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Let's solve the given problems step-by-step.

### Problem 1(a)
**Find the two angles that satisfy the equation:**
\[ 8 \sec x = 3 \tan^2 x + 7 \]
for \(0^\circ < x < 360^\circ\).

**Steps:**
1. Express \(\sec x\) and \(\tan x\) in terms of \(\cos x\) and \(\sin x\).
   \[ \sec x = \frac{1}{\cos x} \]
   \[ \tan x = \frac{\sin x}{\cos x} \]
   \[ \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \]

2. Substitute these into the equation:
   \[ 8 \frac{1}{\cos x} = 3 \frac{\sin^2 x}{\cos^2 x} + 7 \]

3. Simplify by multiplying through by \(\cos^2 x\):
   \[ 8 \cos x = 3 \sin^2 x + 7 \cos^2 x \]

4. Use the Pythagorean identity \(\sin^2 x = 1 - \cos^2 x\):
   \[ 8 \cos x = 3 (1 - \cos^2 x) + 7 \cos^2 x \]
   \[ 8 \cos x = 3 - 3 \cos^2 x + 7 \cos^2 x \]
   \[ 8 \cos x = 3 + 4 \cos^2 x \]

5. Rearrange to form a quadratic equation in \(\cos x\):
   \[ 4 \cos^2 x - 8 \cos x + 3 = 0 \]

6. Solve the quadratic equation using the quadratic formula:
   \[ \cos x = \frac{8 \pm \sqrt{64 - 48}}{8} = \frac{8 \pm 4}{8} \]
   \[ \cos x = \frac{12}{8} = 1.5 \quad (\text{invalid, as } -1 \leq \cos x \leq 1) \]
   \[ \cos x = \frac{4}{8} = 0.5 \]

7. Find the corresponding angles \(x\):
   \[ x = \cos^{-1}(0.5) \]
   \[ x = 60^\circ \text{ and } x = 300^\circ \]

**Answers:**
\[ x = 60^\circ \text{ and } x = 300^\circ \]

### Problem 1(b)
**Find all the angles between \(0\) and \(2\pi\), inclusive, for which:**
\[ 2 \cos 2x = 11 \cos x + 1 \]

**Steps:**
1. Use the double-angle identity for \(\cos 2x\):
   \[ \cos 2x = 2 \cos^2 x - 1 \]
   \[ 2 (2 \cos^2 x - 1) = 11 \cos x + 1 \]
   \[ 4 \cos^2 x - 2 = 11 \cos x + 1 \]

2. Rearrange to form a quadratic equation in \(\cos x\):
   \[ 4 \cos^2 x - 11 \cos x - 3 = 0 \]

3. Solve the quadratic equation using the quadratic formula:
   \[ \cos x = \frac{11 \pm \sqrt{121 + 48}}{8} = \frac{11 \pm \sqrt{169}}{8} \]
   \[ \cos x = \frac{11 \pm 13}{8} \]
   \[ \cos x = 3 \quad (\text{invalid, as } -1 \leq \cos x \leq 1) \]
   \[ \cos x = -\frac{1}{4} \]

4. Find the corresponding angles \(x\) in the interval \(0 \leq x \leq 2\pi\):
   \[ x = \cos^{-1}\left(-\frac{1}{4}\right) \]
   This will give two angles in the range \(0\) to \(2\pi\).

**Answers (to 3 significant figures):**
\[ x = 1.823 \text{ radians and } x = 4.460 \text{ radians} \]

If you have any questions or need further details, feel free to ask!

### Related Questions:
1. What is the definition of secant and tangent in trigonometry?
2. How do you derive the Pythagorean identity for trigonometric functions?
3. What are the general solutions for \(\cos x = a\)?
4. Can you explain the steps to solve a quadratic equation?
5. What are the properties of trigonometric functions over the interval \(0\) to \(360^\circ\)?
6. How do you convert angles from degrees to radians and vice versa?
7. What are the double-angle identities in trigonometry?
8. How do you solve trigonometric equations involving multiple angles?

### Tip:
Always check the range of your trigonometric functions to ensure your solutions are valid.

Learn how to solve trigonometric equations involving secant and tangent functions step-by-step. This problem covers key trigonometric concepts, including the Pythagorean identity and double-angle identities. Suitable for high school students.

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