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# The Maximum value of $4{\sin ^2}x + 3{\cos ^2}x$ is $A - 3$  $B - 4$  $C - 5$  $D - 7$

Last updated date: 18th Jun 2024
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Hint: In order to solve the Trigonometric Numerical , we need to know all the Identities by heart in order to solve the numerical . First step is to identify the type of identity to be used based on the trigonometric function given in the question. Since there are several identities for a single function, the next step is to decide which particular identity would suffice in order to solve the problem. It is also Advisable to Know the Maximum value , Minimum value and value for $\dfrac{\pi }{6}\& {60^ \circ }$ for all trigonometric functions

From the above question we can figure it out that we have to make use of the Trigonometry identities in order to find the correct answer. Since the problem involves $\sin x$ & $\cos x$ we will have to make use of identity
${\sin ^2}\theta + {\cos ^2}\theta = 1..........(1)$
Given : $4{\sin ^2}x + 3{\cos ^2}x$
Restructuring $Equation1$ :
${\cos ^2}\theta = 1 - {\sin ^2}\theta ..........(2)$
Substituting value of ${\cos ^2}\theta$from $Equation2$ into the given problem
We are replacing $\theta$ with $x$ , since the question consists of $x$ . There is no compulsion on using $\theta$ with Trigonometric functions.
$\Rightarrow 4{\sin ^2}x + 3 \times (1 - {\sin ^2}x)$
On simplifying the above equation and on opening the brackets we will get the following equation
$\Rightarrow 4{\sin ^2}x + 3 - 3{\sin ^2}x........(3)$
Further Simplifying the $Equation3$ in order to bring it to non-reducible form
$\Rightarrow {\sin ^2}x + 3.......(4)$
Now remembering the properties of $\sin x$ we can say that the value of $\sin x$ ranges from $- 1$ to $1$ .
Thus the maximum value of $\sin x$ will always be $1$ . Thus the maximum value of ${\sin ^2}x$ will be $1$ .
$\therefore$ Substituting Maximum Value of ${\sin ^2}x$ $Equation4$
Final answer on substituting would be $4$ .
Thus the maximum value of $4{\sin ^2}x + 3{\cos ^2}x$ is $4$ .
From the given options, the correct option is $OptionB$ which has a value of $4$ .
So, the correct answer is “Option B”.

Note: It is also Advisable to Know the Maximum value , Minimum value and value $\dfrac{\pi }{6}\& {60^ \circ }$ for all trigonometric functions. This is because in almost all the sums these are the standard angles based on which the questions are asked. Apart from these , memorizing the formulas and the relationship between various trigonometric functions is extremely vital.