Question

An extreme value of $1+4\sin \theta +3\cos \theta$ is

• A. $-3$
• B. $-4$
• C. $5$
• D. $6$

Answer 1

Answer: (D)

$6$

As we know that

The range of $a\cos x+b\sin x+c$ is $[c-\sqrt{a^2+b^2},c+\sqrt{a^2+b^2}]$

Here $a=3,b=4,c=1$

So the range of $3\cos \theta +4\sin \theta +1$ is $[1-\sqrt{3^2+4^2},1+\sqrt{3^2+4^2}]$

So the extreme value is $1+\sqrt{3^2+4^2}=1+5=6$