Answer
438k+ views
Hint:Using the equation ${\sin ^2}x + {\cos ^2}x = 1$ change the equation into a quadratic equation in sine or cosine function. Now find the discriminant of the quadratic equation to find out the characteristics of the solution of the equation. Since a sine or cosine function can have imaginary value, so the discriminant must be non-negative for real solutions.
Complete step-by-step answer:
We know that: ${\sin ^2}x + {\cos ^2}x = 1 \Rightarrow \cos x = \sqrt {1 - {{\sin }^2}x} $
Using the above relationship, we can express $\cos x$ in the form of $\sin x$. So, now our given equation will become:
$ \Rightarrow \sqrt 3 \sin x + \sqrt {1 - {{\sin }^2}x} = 4$
$ \Rightarrow \sqrt {1 - {{\sin }^2}x} = 4 - \sqrt 3 \sin x$
Now squaring both the sides of the above equation, we have a simpler equation
$ \Rightarrow 1 - {\sin ^2}x = {\left( {4 - \sqrt 3 \sin x} \right)^2}$
Let us now expand this by using ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ and simplifying it further
$ \Rightarrow 1 - {\sin ^2}x = {4^2} + {\left( {\sqrt 3 \sin x} \right)^2} - 2 \times 4 \times \sqrt 3 \sin x$
$ \Rightarrow 1 - {\sin ^2}x = 16 + 3{\sin ^2}x - 8\sqrt 3 \sin x$
$ \Rightarrow 4{\sin ^2}x - 8\sqrt 3 \sin x + 15 = 0………...(i)$
Here we got a quadratic equation in $\sin x$ and we can check for the roots for this equation.
It is important to understand a few things about quadratic equations first. If we have an equation of the form $a{x^2} + bx + c = 0$. We can use the expression,${b^2} - 4ac$, also known as the discriminant to determine the number of roots of solutions in a quadratic equation. There are three cases:
${b^2} - 4ac < 0$: The equation has zero real solutions. The graph does not cross the x-axis.
${b^2} - 4ac = 0$: The equation has one real solution. The graph touches the x-axis at one point.
${b^2} - 4ac > 0$: The equation has two real solutions. The graph crosses through the x-axis at two points.
Now let us use this same concept for solving this problem.
For our previous equation $(i)$, ${b^2} - 4ac = {\left( {8\sqrt 3 } \right)^2} - 4 \times 4 \times 15 = 64 \times 3 - 16 \times 15 = 192 - 240 = - 48 < 0$
So here the discriminant is negative, which signifies that this quadratic equation has no real solutions, which means there is no real value of $\sin x$ to satisfy this equation.
Therefore, we can conclude that the equation $\sqrt 3 \sin x + \cos x = 4$ does not have any solution.
So, the correct answer is “Option B”.
Note:An alternative approach can be taken in changing the given equation into a quadratic equation in cosine function. You can also approach it by changing the given equation using \[\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b\] or \[\cos \left( {a - b} \right) = \cos a\cos b + \sin a\sin b\] .
Complete step-by-step answer:
We know that: ${\sin ^2}x + {\cos ^2}x = 1 \Rightarrow \cos x = \sqrt {1 - {{\sin }^2}x} $
Using the above relationship, we can express $\cos x$ in the form of $\sin x$. So, now our given equation will become:
$ \Rightarrow \sqrt 3 \sin x + \sqrt {1 - {{\sin }^2}x} = 4$
$ \Rightarrow \sqrt {1 - {{\sin }^2}x} = 4 - \sqrt 3 \sin x$
Now squaring both the sides of the above equation, we have a simpler equation
$ \Rightarrow 1 - {\sin ^2}x = {\left( {4 - \sqrt 3 \sin x} \right)^2}$
Let us now expand this by using ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ and simplifying it further
$ \Rightarrow 1 - {\sin ^2}x = {4^2} + {\left( {\sqrt 3 \sin x} \right)^2} - 2 \times 4 \times \sqrt 3 \sin x$
$ \Rightarrow 1 - {\sin ^2}x = 16 + 3{\sin ^2}x - 8\sqrt 3 \sin x$
$ \Rightarrow 4{\sin ^2}x - 8\sqrt 3 \sin x + 15 = 0………...(i)$
Here we got a quadratic equation in $\sin x$ and we can check for the roots for this equation.
It is important to understand a few things about quadratic equations first. If we have an equation of the form $a{x^2} + bx + c = 0$. We can use the expression,${b^2} - 4ac$, also known as the discriminant to determine the number of roots of solutions in a quadratic equation. There are three cases:
${b^2} - 4ac < 0$: The equation has zero real solutions. The graph does not cross the x-axis.
${b^2} - 4ac = 0$: The equation has one real solution. The graph touches the x-axis at one point.
${b^2} - 4ac > 0$: The equation has two real solutions. The graph crosses through the x-axis at two points.
Now let us use this same concept for solving this problem.
For our previous equation $(i)$, ${b^2} - 4ac = {\left( {8\sqrt 3 } \right)^2} - 4 \times 4 \times 15 = 64 \times 3 - 16 \times 15 = 192 - 240 = - 48 < 0$
So here the discriminant is negative, which signifies that this quadratic equation has no real solutions, which means there is no real value of $\sin x$ to satisfy this equation.
Therefore, we can conclude that the equation $\sqrt 3 \sin x + \cos x = 4$ does not have any solution.
So, the correct answer is “Option B”.
Note:An alternative approach can be taken in changing the given equation into a quadratic equation in cosine function. You can also approach it by changing the given equation using \[\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b\] or \[\cos \left( {a - b} \right) = \cos a\cos b + \sin a\sin b\] .
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)