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Let's work through the questions in the image:

### 11. **State the exact value of \( \cos(30^\circ) \):**
The exact value of \( \cos(30^\circ) \) is:
\[
\cos(30^\circ) = \frac{\sqrt{3}}{2}
\]

### 12. **If \( 0^\circ \leq x \leq 360^\circ \), solve \( \cos x = -0.438 \):**
To solve for \( x \) where \( \cos x = -0.438 \), follow these steps:
1. First, find the reference angle:
   \[
   \cos^{-1}(0.438) \approx 64^\circ
   \]
   This is the reference angle in the first quadrant.

2. Since cosine is negative, the solutions must be in the second and third quadrants.

   - In the second quadrant:
     \[
     x = 180^\circ - 64^\circ = 116^\circ
     \]
   - In the third quadrant:
     \[
     x = 180^\circ + 64^\circ = 244^\circ
     \]
Thus, the two solutions are:
\[
x = 116^\circ \quad \text{and} \quad x = 244^\circ
\]

### 12 (Second part). **Verify that \( \cos x = \sec x - \frac{\sin x}{\cot x} \):**
We need to verify this identity:
\[
\cos x = \sec x - \frac{\sin x}{\cot x}
\]

1. Start by expressing the terms on the right-hand side:
   \[
   \sec x = \frac{1}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}
   \]
   Therefore:
   \[
   \frac{\sin x}{\cot x} = \frac{\sin x}{\frac{\cos x}{\sin x}} = \frac{\sin^2 x}{\cos x}
   \]

2. Substituting into the equation:
   \[
   \sec x - \frac{\sin x}{\cot x} = \frac{1}{\cos x} - \frac{\sin^2 x}{\cos x}
   \]

3. Factor out \( \frac{1}{\cos x} \):
   \[
   \frac{1 - \sin^2 x}{\cos x}
   \]

4. Use the identity \( 1 - \sin^2 x = \cos^2 x \):
   \[
   \frac{\cos^2 x}{\cos x} = \cos x
   \]
Thus, both sides are equal, verifying the identity.

### 13. **Sketch the graph of \( y = 2\sin\left(\frac{3}{2}x\right) + 1 \), for \( 0^\circ \leq x \leq 360^\circ \):**
This is a sinusoidal function with:
- Amplitude: \( 2 \)
- Vertical shift: \( +1 \)
- Period: The period is \( \frac{360^\circ}{\frac{3}{2}} = 240^\circ \).

To sketch the graph:
1. The function oscillates between \( 1 - 2 = -1 \) and \( 1 + 2 = 3 \).
2. The sine wave completes one cycle in \( 240^\circ \), so it will complete 1.5 cycles in \( 360^\circ \).

### Domain and Range:
- **Domain**: \( [0^\circ, 360^\circ] \)
- **Range**: \( [-1, 3] \)

Would you like to go deeper into any of these explanations or solve additional questions related to this topic?

### Here are 5 related questions to explore:
1. What is the general solution for \( \cos x = -0.438 \) over all angles?
2. How would you derive the period of a sinusoidal function like \( y = 2\sin\left(\frac{3}{2}x\right) + 1 \)?
3. Can you explain the transformation effects of adding and multiplying constants to trigonometric functions?
4. How would you solve \( \sin x = 0.5 \) over the interval \( [0^\circ, 360^\circ] \)?
5. What is the relationship between \( \sin x \) and \( \cos x \) in the unit circle?

### Tip:
When solving trigonometric equations, always consider both the positive and negative solutions in the relevant quadrants, and use the unit circle to visualize the angles.

State the exact value of cos(30°). Solve for x where 0° ≤ x ≤ 360°, given that cos x = -0.438. Verify that cos x = sec x - (sin x / cot x). Sketch the graph of y = 2sin(3/2)x + 1 for 0° ≤ x ≤ 360°. State the domain and range for the graph.

Explore solutions for trigonometric problems, including the exact value of cos(30°), solving cos x = -0.438 for 0° ≤ x ≤ 360°, verifying trigonometric identities, and graphing y = 2sin(3/2)x + 1. Suitable for high school students, this covers trigonometric equations, identities, and graphing sinusoidal functions.

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