Question

The maximum value of $3\cos\theta - 4\sin\theta$ is

• A. 3
• B. 4
• C. 5
• D. None of these

Answer 1

Let $3 = r\cos\alpha,4 = r\sin\alpha,$ so $r = 5$

$f(\theta) = r.(\cos\alpha\cos\theta + \sin\alpha\sin\theta) = 5.\cos(\theta - \alpha)$

$\therefore$ The maximum value of $f(\theta) = 5.1 = 5.$

{Since the maximum value of $\cos(\theta - \alpha) = 1$}.

Aliter : As we know that, the maximum value of

$a\sin\theta + b\cos\theta$is $+ \sqrt{a^{2} + b^{2}}$ and the minimum value is

$- \sqrt{a^{2} + b^{2}}$. Therefore, the maximum value is

$(3\cos\theta + 4\sin\theta) = + \sqrt{3^{2} + ( - 4)^{2}} = 5$ and the minimum value is – 5.