Question

The length of the longest interval, in which $f(x)=3\sin x-4{{\sin }^{3}}x$ is increasing, is
A. $\dfrac{\pi }{3}$
B. $\dfrac{\pi }{2}$
C. $\dfrac{3\pi }{2}$
D. $\pi$

Answer 1

For solving this question, first we have to observe the given function and then use some trigonometric formula to make it simpler and after that find the interval in which the given trigonometric function is increasing after finding the interval, calculate the length of the longest interval.

Complete step by step answer:
Trigonometry can be defined as a study of the relationship of angles, lengths and heights. There are total six types of different functions in trigonometry: Sine $(\sin )$, Cosine $(\cos )$, Secant $(\sec )$, Cosecant $(\cos ec)$ , Tangent $(\tan )$ and Cotangent $(\cot )$. Basically these six types of trigonometric functions define the relationship between the different sides of a right angle triangle.
Sine is defined as the ratio of opposite side to the hypotenuse i.e.
$\sin \theta =\dfrac{opposite}{hypotenuse}$
Cosine is defined as the ratio of adjacent side to the hypotenuse i.e.
$\cos \theta =\dfrac{adjacent}{hypotenuse}$
Tangent can be defined as the ratio of opposite side to the adjacent side i.e.
$\tan \theta =\dfrac{opposite}{adjacent}$
Cotangent is the reciprocal of the tangent. So it is the ratio of adjacent side to the opposite side as:
$\cot \theta =\dfrac{adjacent}{opposite}$
Cosecant is the reciprocal of sine. So it can be defined as the ratio of hypotenuse to the opposite side.
$\co sec\theta =\dfrac{hypotenuse}{opposite}$
Secant is the reciprocal of the cosine. So it can be defined as the ratio of hypotenuse to adjacent side.
$\sec \theta =\dfrac{hypotenuse}{adjacent}$

Sine can be defined as the ratio of the opposite side of a right triangle to its hypotenuse. The sine function has values positive or negative upon the quadrants. Sine is positive in the first and second quadrants while it is negative for the third and fourth quadrants.
We have given in the question: $f(x)=3\sin x-4{{\sin }^{3}}x$
As there is a formula, $\sin 3x=3\sin x-4{{\sin }^{3}}x$
Now by using above formula, our function becomes as:
$f(x)=\sin 3x$
As $\sin x$ increases in the interval $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$
Therefore, for $\sin 3x$ :
$-\dfrac{\pi }{2}\le 3x\le \dfrac{\pi }{2}$
$\Rightarrow -\dfrac{\pi }{6}\le x\le \dfrac{\pi }{6}$
Now, we have to calculate the length of the longest interval. It is calculated by subtracting the upper range of interval and the lower range of interval.
So, the length of the interval is given as:
$=\left[ \dfrac{\pi }{6}-\left( -\dfrac{\pi }{6} \right) \right]$
$\Rightarrow \dfrac{2\pi }{6}$
$\Rightarrow \dfrac{\pi }{3}$
Therefore, the length of the longest interval in which $f(x)$ increases is $\dfrac{\pi }{3}$

So, the correct answer is “Option A”.

Note: Length of sine wave is called wavelength. Sine waves also have other properties such as frequency, amplitude, phase and speed. The frequency of a sine wave is the number of oscillations a wave has in a certain period of time, for example: one second.