Question

The maximum value of \(3 \cos \theta + 4 \sin \theta\) is

• A. 2
• B. 4
• C. 5
• D. 1

Hint:

For any expression of the form \(a \sin x + b \cos x\), immediately calculate \(\sqrt{a^2+b^2}\). This value is the maximum value (amplitude) of the resulting single trigonometric function. This is a very frequent question type, so knowing the formula for the maximum and minimum values is a must.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the maximum possible value of the expression \(3 \cos \theta + 4 \sin \theta\).

Step 2: Key Formula or Approach:
An expression of the form \(a \cos \theta + b \sin \theta\) can be converted into the form \(R \cos(\theta - \alpha)\) or \(R \sin(\theta + \beta)\).
The range of such an expression is \([-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]\).
The maximum value is \(\sqrt{a^2+b^2}\) and the minimum value is \(-\sqrt{a^2+b^2}\).

Step 3: Detailed Explanation:
The given expression is \(3 \cos \theta + 4 \sin \theta\).
This is in the form \(a \cos \theta + b \sin \theta\) with \(a=3\) and \(b=4\).
Using the formula, the maximum value is \(\sqrt{a^2+b^2}\).
Calculate the value:
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \]
The minimum value would be -5. The entire expression ranges from -5 to 5.
Therefore, the maximum value is 5.

Derivation (for understanding):
Let \(3 = R \cos \alpha\) and \(4 = R \sin \alpha\).
Squaring and adding these gives:
\(3^2 + 4^2 = R^2 \cos^2 \alpha + R^2 \sin^2 \alpha\)
\(25 = R^2(\cos^2 \alpha + \sin^2 \alpha)\)
\(25 = R^2 \implies R = 5\) (since R is a magnitude).
The original expression becomes:
\(R \cos \alpha \cos \theta + R \sin \alpha \sin \theta = R(\cos \theta \cos \alpha + \sin \theta \sin \alpha)\)
Using the identity \(\cos(A-B) = \cos A \cos B + \sin A \sin B\), this is:
\(R \cos(\theta - \alpha) = 5 \cos(\theta - \alpha)\).
Since the maximum value of any cosine function is 1, the maximum value of \(5 \cos(\theta - \alpha)\) is \(5 \times 1 = 5\).

Step 4: Final Answer:
The maximum value of the expression \(3 \cos \theta + 4 \sin \theta\) is 5.