If rcos(A-1.18), find the greatest value of 5cosA+12sinA and value of A for which it occurs (0≤A≤2π)

We are asked to find the greatest value of the expression \( 5\cos A + 12\sin A \) and the value of \( A \) at which this maximum occurs, given the range \( 0 \leq A \leq 2\pi \).

This is a classic maximization problem involving trigonometric functions, and we can solve it by expressing \( 5\cos A + 12\sin A \) in a simpler form. The general strategy involves rewriting the expression in the form \( R\cos(A - \alpha) \), where \( R \) and \( \alpha \) are constants.

### Step 1: Express \( 5\cos A + 12\sin A \) in the form \( R\cos(A - \alpha) \)
To do this, we use the identity:
\[
R\cos(A - \alpha) = R\cos \alpha \cos A + R\sin \alpha \sin A
\]
By comparing this with \( 5\cos A + 12\sin A \), we can equate the coefficients:
\[
R\cos \alpha = 5 \quad \text{and} \quad R\sin \alpha = 12
\]

### Step 2: Solve for \( R \) and \( \alpha \)
To find \( R \), we use the Pythagorean identity:
\[
R = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
So, \( R = 13 \).

Next, we calculate \( \alpha \):
\[
\tan \alpha = \frac{12}{5}
\]
Thus, \( \alpha = \tan^{-1} \left( \frac{12}{5} \right) \).

### Step 3: Find the maximum value
The expression \( 5\cos A + 12\sin A \) is now written as:
\[
13\cos(A - \alpha)
\]
The maximum value of \( \cos(A - \alpha) \) is 1, so the maximum value of \( 13\cos(A - \alpha) \) is \( 13 \times 1 = 13 \).

### Step 4: Determine the value of \( A \)
The maximum occurs when \( \cos(A - \alpha) = 1 \), which happens when \( A - \alpha = 0 \), or \( A = \alpha \). Therefore, \( A = \tan^{-1} \left( \frac{12}{5} \right) \).

Using a calculator:
\[
\alpha = \tan^{-1} \left( \frac{12}{5} \right) \approx 1.176 \, \text{radians}
\]

Thus, the maximum value of \( 5\cos A + 12\sin A \) is 13, and it occurs when \( A \approx 1.176 \, \text{radians} \).

### Summary:
- The maximum value is \( 13 \).
- This maximum occurs when \( A \approx 1.176 \, \text{radians} \).

Would you like more details or clarifications on this solution? Here are some related questions:

1. How do we derive the identity \( R\cos(A - \alpha) \)?
2. Why is the maximum value of a cosine function always 1?
3. Can the method of expressing as \( R\cos(A - \alpha) \) be applied to other trigonometric expressions?
4. What is the geometric interpretation of \( 5\cos A + 12\sin A \)?
5. How would the solution change if the expression was \( 12\cos A + 5\sin A \)?

**Tip:** Rewriting trigonometric expressions in terms of a single cosine function is a powerful technique for finding maximum and minimum values.

Explore how to find the greatest value of the expression 5cosA + 12sinA and the corresponding value of A within the range 0≤A≤2π. Learn key trigonometric techniques such as rewriting expressions in the form Rcos(A-α), calculating R using the Pythagorean identity, and determining the maximum value of cosine functions. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation