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To express the function \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\), we can use the following identity:

\[
R \cos(\theta - \alpha) = R \left(\cos \alpha \cos \theta + \sin \alpha \sin \theta\right)
\]

We want to match this with the given expression \(5 \sin \theta + 12 \cos \theta\).

### Steps:

1. **Compare the coefficients**: 
   - We can write:
     \[
     5 \sin \theta + 12 \cos \theta = R \sin \alpha \sin \theta + R \cos \alpha \cos \theta
     \]
   - By comparing coefficients of \(\sin \theta\) and \(\cos \theta\), we get:
     \[
     R \sin \alpha = 5 \quad \text{and} \quad R \cos \alpha = 12
     \]

2. **Find \(R\)**: 
   - Use the Pythagorean identity:
     \[
     R = \sqrt{(R \sin \alpha)^2 + (R \cos \alpha)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
     \]

3. **Find \(\alpha\)**:
   - Use the definitions of \(\sin \alpha\) and \(\cos \alpha\):
     \[
     \sin \alpha = \frac{5}{13}, \quad \cos \alpha = \frac{12}{13}
     \]
   - So, \(\alpha\) is given by:
     \[
     \alpha = \tan^{-1} \left(\frac{5}{12}\right)
     \]

Thus, the expression can be written as:

\[
5 \sin \theta + 12 \cos \theta = 13 \cos\left(\theta - \alpha\right)
\]

where \(\alpha = \tan^{-1} \left(\frac{5}{12}\right)\).

Would you like any further explanations or have additional questions?

### Relative Questions:
1. How do you derive the identity for \(R \cos(\theta - \alpha)\)?
2. What are the steps to find \(R\) and \(\alpha\) in such problems?
3. How can the tangent function be used to determine the angle in trigonometric identities?
4. What is the significance of expressing trigonometric functions in the form \(R \cos(\theta - \alpha)\)?
5. How would the problem change if the coefficients of \(\sin \theta\) and \(\cos \theta\) were different?
6. What is the importance of the angle \(\alpha\) being in the first quadrant (\(0 < \alpha < \frac{\pi}{2}\))?
7. How can this approach be extended to functions involving other trigonometric expressions?
8. What are some applications of expressing trigonometric sums in this form?

### Tip:
Always ensure that the values of \(\sin \alpha\) and \(\cos \alpha\) correspond to a valid angle \(\alpha\) in the correct quadrant when using trigonometric identities.

Learn how to express the trigonometric sum 5 sin(theta) + 12 cos(theta) in the form R cos(theta - alpha) using trigonometric identities. This detailed solution covers the derivation of the identity, steps to find R and alpha, and the significance of expressing trigonometric functions in this form. Suitable for advanced high school students studying trigonometry.

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