Question

The minimum value of $3\cos x+4\sin x+5$ is

• A. $5$
• B. $9$
• C. $7$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for the correct option:

Find the minimum value of the given trigonometric function.

A trigonometric function $3\cos x+4\sin x+5$ is given.

We know that, the minimum value of the function $a\cos x+b\sin x$ is given by $-\sqrt{a^2+b^2}$.

Therefore, the minimum value of the given function is given by:

$$\begin{gathered} -\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}+5=-\sqrt{9+16}+5 \\ \Rightarrow -\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}+5=-\sqrt{25}+5 \\ \Rightarrow -\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}+5=-5+5 \\ \Rightarrow -\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}+5=0 \\ \Rightarrow -5+5=0 \end{gathered}$$

Therefore, the minimum value of $3\cos x+4\sin x+5$ is $0$.

Hence, the option $D$ is the correct answer.