Question

The maximum value of the function \(a \sin x+b \cos x\) is

• A. \(\sqrt{a^{2}+b^{2}}\)
• B. \(\sqrt{a^{2}-b^{2}}\)
• C. \(\mathrm{a}^{2}+\mathrm{b}^{2}\)
• D. \(\mathrm{a}^{2}-\mathrm{b}^{2}\)

Hint:

Write a sin x + b cos x as a single sine with an amplitude.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Any expression \(a\sin x+b\cos x\) can be written as \(R\sin(x+\alpha)\).

Step 2: Find R
Take \(a=R\cos\alpha\) and \(b=R\sin\alpha\). Then \(R^2=a^2+b^2\), so \(R=\sqrt{a^2+b^2}\).

Step 3: Maximum
Since \(\sin(x+\alpha)\) has maximum \(1\), the maximum value is \(\sqrt{a^2+b^2}\), option (A).

Final Answer:
The maximum value is \(\sqrt{a^2+b^2}\). \[ \boxed{\text{(A) }\sqrt{a^{2}+b^{2}}} \]