Answer
384.6k+ views
Hint:As we know that the above given question is related to trigonometric expression, sine and cosine are trigonometric ratios. Here we have to find the value of $f(x)$, so first of all we have to solve and simplify the value. We can convert the equation into basic trigonometric equations by applying the trigonometric identities.
Complete step by step solution:
As per the given question we have to solve the expression $f(x) = \dfrac{{\sin 3x}}{{\sin x}}$. We know that a trigonometric identity for $\sin 3x$ which is $3\sin x - 4{\sin ^3}x$, now by substituting this value we will expand and we get, $f(x) = \dfrac{{3\sin x - 4{{\sin }^3}x}}{{\sin x}}$. We can take the common $\sin x$ out and it gives,
$\dfrac{{\sin x(3 - 4{{\sin }^2}x)}}{{\sin x}} = 3 - 4{\sin ^2}x$. Now let us assume $3 - 4{\sin ^2}x = y$, so we get: ${\sin ^2}x = \dfrac{{3 - y}}{4}$.
Since we can say that $0 \leqslant \dfrac{{3 - y}}{4} \leqslant 1$. Now solving this by inequality :
$ = 0 \leqslant 3 - y \leqslant 4 \Rightarrow - 3 \leqslant - y \leqslant 1$.
Now we can interchange the values but by keeping the signs not changed, $ - 1 \leqslant y \leqslant 3$.
So we can write it as $y \in [ - 1,3]$.
Hence the correct option is (d) $[ - 1,3]$.
Note: Before solving this kind of question we should have the proper knowledge of all trigonometric ratios, identities and their formulas. To solve this trigonometric expression we should also have the proper knowledge of the inequality, and then we should solve it by avoiding mistakes and taking care of positive and negative signs.
Complete step by step solution:
As per the given question we have to solve the expression $f(x) = \dfrac{{\sin 3x}}{{\sin x}}$. We know that a trigonometric identity for $\sin 3x$ which is $3\sin x - 4{\sin ^3}x$, now by substituting this value we will expand and we get, $f(x) = \dfrac{{3\sin x - 4{{\sin }^3}x}}{{\sin x}}$. We can take the common $\sin x$ out and it gives,
$\dfrac{{\sin x(3 - 4{{\sin }^2}x)}}{{\sin x}} = 3 - 4{\sin ^2}x$. Now let us assume $3 - 4{\sin ^2}x = y$, so we get: ${\sin ^2}x = \dfrac{{3 - y}}{4}$.
Since we can say that $0 \leqslant \dfrac{{3 - y}}{4} \leqslant 1$. Now solving this by inequality :
$ = 0 \leqslant 3 - y \leqslant 4 \Rightarrow - 3 \leqslant - y \leqslant 1$.
Now we can interchange the values but by keeping the signs not changed, $ - 1 \leqslant y \leqslant 3$.
So we can write it as $y \in [ - 1,3]$.
Hence the correct option is (d) $[ - 1,3]$.
Note: Before solving this kind of question we should have the proper knowledge of all trigonometric ratios, identities and their formulas. To solve this trigonometric expression we should also have the proper knowledge of the inequality, and then we should solve it by avoiding mistakes and taking care of positive and negative signs.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 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)
Give 10 examples for herbs , shrubs , climbers , creepers
![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)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english 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)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)