Question

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

Answer 1

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

Complete step-by-step answer :
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.