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Maximum value of the expression \[2\sin x+4\cos x+3\] is
A. \[2\sqrt{5}+3\]
B. \[2\sqrt{5}-3\]
C. \[\sqrt{5}+3\]
D. None of these.

Last updated date: 23rd Jul 2024
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Hint: Use trigonometric R-method to find the maxima of a given expression. Find the formula of the R-method and compare it with the expression given. Then solve it to get the maximum value.

“Complete step-by-step answer:”
We have been given the expression, \[2\sin x+4\cos x+3\].
So, let us put, \[f\left( x \right)=2\sin x+4\cos x+3\].
The maximum value of \[a\sin x+b\cos x\]is equal to \[\sqrt{{{a}^{2}}+{{b}^{2}}}\].
This equation \[\left( a\sin x+b\cos x \right)\] is similar to the expression, \[2\sin x+4\cos x+3\].
Let us use the Trigonometric R-method to solve this expression.
The R-method is used to find the extrema (maxima and minimum) of combinations of trigonometric function.
Let us consider, \[y=A\sin x+b\sin x\].
Thus by using the R-formula, let us express y as, \[y=a\sin x+b\cos x=R\sin \left( x+\theta \right)\].
For maximum value of y, \[\sin \left( x+\theta \right)=1\].
\[\therefore \]Maximum value of, \[y=R\left( 1 \right)=R\].
We know, \[\sin \left( a+b \right)=\sin a\cos b+\cos a\sin b\].
We need to find the values of R.
\[a\sin x+b\cos x=\left( R\cos \theta \right)\sin x+\left( R\sin \theta \right)\cos x\]
By comparing we can see that, \[a=R\cos \theta \] and \[b=R\sin \theta \].
  & \therefore \dfrac{b}{a}=\dfrac{R\sin \theta }{R\cos \theta }=\tan \theta \\
 & \therefore \tan \theta =\dfrac{b}{a} \\
 & \Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right) \\
\[a=R\cos \theta \]
  & R=\dfrac{a}{\cos \theta }=\dfrac{a}{\cos {{\tan }^{-1}}\left( \dfrac{b}{a} \right)}=\dfrac{a}{\dfrac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}}} \\
 & R=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
 & \therefore a\sin x+b\cos x=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
So, value of a = 2 and b = 4.
  & f\left( x \right)=2\sin x+4\cos x+3 \\
 & f\left( x \right)=\sqrt{{{a}^{2}}+{{b}^{2}}}+3 \\
 & f\left( x \right)=\sqrt{{{2}^{2}}+{{4}^{2}}}+3 \\
 & \therefore f\left( x \right)=\sqrt{20}+3=\sqrt{2\times 2\times 5}+3 \\
 & f\left( x \right)=2\sqrt{5}+3 \\
Therefore, the maximum value of f (x) becomes, \[\sqrt{20}+3=2\sqrt{5}+3\].
Hence, option (a) is the correct answer.

Note: Maxima and Minima are important concepts in trigonometry. Here we did the proof for finding maximum value. You can simply apply the expression in the given function f (x) as they are similar.
If \[f\left( x \right)=2\sin x+4\cos x\], then \[f\left( x \right)=\sqrt{{{a}^{2}}+{{b}^{2}}}=\sqrt{{{2}^{2}}+{{4}^{2}}}=\sqrt{20}\].
The maximum value of f (x) is \[\sqrt{20}\].