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 θ + 4 sin θ.

Step 2: Key Formula or Approach (Alternate Method):
Use the formula for maximum of a cos θ + b sin θ, which is √(a² + b²). This comes from rewriting as R cos(θ - α).

Step 3: Detailed Explanation:
Given expression: 3 cos θ + 4 sin θ. Here a = 3, b = 4. Maximum value formula: R = √(a² + b²). R = √(3² + 4²) = √(9 + 16) = √25 = 5. This is because we can write 3 cos θ + 4 sin θ = 5 cos(θ - α) where tan α = 4/3. The maximum of cosine function is 1, so max = 5.

Step 4: Final Answer:
The maximum value of the expression 3 cos θ + 4 sin θ is 5.