Find the maximum value of 10sin x + 24cos x.

To find the maximum value of the expression \( 10\sin x + 24\cos x \), we can proceed as follows:

1. **Express the function in the form \( R\sin(x + \alpha) \):**
   \[
   10\sin x + 24\cos x = R\sin(x + \alpha)
   \]
   where \( R = \sqrt{10^2 + 24^2} \) and \( \tan \alpha = \frac{24}{10} \).

2. **Calculate \( R \):**
   \[
   R = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26
   \]

3. **Maximum value of \( R\sin(x + \alpha) \):**
   The maximum value of \( \sin(x + \alpha) \) is 1, so the maximum value of \( R\sin(x + \alpha) \) is \( R \). Thus,
   \[
   \text{Maximum value} = 26
   \]

Therefore, the maximum value of \( 10\sin x + 24\cos x \) is **26**.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. What is the minimum value of \( 10\sin x + 24\cos x \)?
2. How do you find the angle \( \alpha \) in the expression \( R\sin(x + \alpha) \)?
3. What is the general method to find the maximum value of \( a\sin x + b\cos x \)?
4. Can this method be used to find the maximum value of other trigonometric functions?
5. How would the solution change if the coefficients of \( \sin x \) and \( \cos x \) were different?

**Tip:** The method of expressing the function as \( R\sin(x + \alpha) \) can be used for any linear combination of sine and cosine to easily find the maximum and minimum values.

Discover how to find the maximum value of the expression 10sin x + 24cos x using trigonometric identities and formulas. Learn the step-by-step method and apply it to solve similar problems.

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