Answer

Verified

341.1k+ 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.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Why is the adrenaline hormone called fight or flight class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between lanthanoids and actinoids class 12 chemistry CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples of unisexual and bisexual flowers

Open circulatory system is present in I Arthropods class 12 biology CBSE

Name the highest peak of the Indian Himalayas class 8 social science CBSE