Answer
Verified
456.6k+ 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
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
What percentage of the solar systems mass is found class 8 physics CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE