Find the maximum value of 8sin theta - 15cos theta + 1

To find the maximum value of the expression \(8\sin\theta - 15\cos\theta + 1\), we can approach it using trigonometric identities and properties.

### Step 1: Express in a single trigonometric function
First, we rewrite the expression \(8\sin\theta - 15\cos\theta\) in the form of \(R\sin(\theta + \alpha)\), where \(R\) and \(\alpha\) are constants to be determined. The identity is:
\[
R\sin(\theta + \alpha) = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha
\]

Comparing this with \(8\sin\theta - 15\cos\theta\), we have:
\[
R\cos\alpha = 8 \quad \text{and} \quad R\sin\alpha = -15
\]

### Step 2: Find \(R\) and \(\alpha\)
The magnitude \(R\) is given by:
\[
R = \sqrt{(8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\]

The angle \(\alpha\) is found using:
\[
\cos\alpha = \frac{8}{17}, \quad \sin\alpha = \frac{-15}{17}
\]

So, the expression becomes:
\[
8\sin\theta - 15\cos\theta = 17\sin(\theta + \alpha)
\]

### Step 3: Find the maximum value
The maximum value of \(\sin(\theta + \alpha)\) is 1. Therefore, the maximum value of \(17\sin(\theta + \alpha)\) is \(17\).

### Step 4: Add the constant
The given expression is \(8\sin\theta - 15\cos\theta + 1\). Adding the constant 1 to the maximum value of \(17\), we get:
\[
\text{Maximum value} = 17 + 1 = 18
\]

### Final Answer
The maximum value of the expression \(8\sin\theta - 15\cos\theta + 1\) is \(\boxed{18}\).

Would you like more details or have any questions?

Here are some related questions:
1. How would you find the minimum value of \(8\sin\theta - 15\cos\theta + 1\)?
2. Can you explain how to find the angle \(\alpha\) explicitly?
3. What if the expression was \(8\sin\theta + 15\cos\theta + 1\), how would the maximum value change?
4. How does the value of \(R\) influence the amplitude of the function?
5. What are the implications of adding a constant to a trigonometric expression in terms of graph transformations?

**Tip:** When dealing with expressions of the form \(a\sin\theta + b\cos\theta\), rewriting it as \(R\sin(\theta + \alpha)\) simplifies finding the extrema.

Learn how to find the maximum value of the expression 8sin theta - 15cos theta + 1 using trigonometric identities and properties. Understand the process step-by-step with detailed explanations and formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation