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Question: The value of x for the maximum value of \(\sqrt{3}\cos x + \sin x\)is...

The value of x for the maximum value of 3cos⁡x+sin⁡x\sqrt{3}\cos x + \sin xis

A

30°

B

45°

C

60°

D

90°

Answer

30°

Explanation

Solution

Let f(x)=3cos⁡x+sin⁡xf(x) = \sqrt{3}\cos x + \sin x

⇒f(x)=2(32cos⁡x+12sin⁡x)=2sin⁡(x+π3)\Rightarrow f(x) = 2\left( \frac{\sqrt{3}}{2}\cos x + \frac{1}{2}\sin x \right) = 2\sin\left( x + \frac{\pi}{3} \right)

But −1≤sin⁡(x+π3)≤1- 1 \leq \sin\left( x + \frac{\pi}{3} \right) \leq 1

Hence, f(x)f(x) is maximum, if x+π3=90∘⇒x=30∘x + \frac{\pi}{3} = 90{^\circ} \Rightarrow x = 30{^\circ}.