Question

How do you solve $3{\csc ^2}x = 4$?

Answer 1

In order to solve this question ,transpose everything from left-Hand side to right-hand side except ${\csc ^2}x$by dividing both sides of the equation by 3 and then put ${\csc ^2}x = \dfrac{1}{{{{\sin }^2}x}}$,take reciprocal of the equation and determine the angle whose sine is equivalent to $\pm \dfrac{{\sqrt 3 }}{2}$to get the set of desired solutions.

Complete step by step solution:
We are given a trigonometric expression$3{\csc ^2}x = 4$
$\Rightarrow 3{\csc ^2}x = 4 \\\ \Rightarrow {\csc ^2}x = \dfrac{4}{3} \\\$
As we know that in trigonometry ${\csc ^2}x = \dfrac{1}{{{{\sin }^2}x}}$,putting the above
expression
$\Rightarrow \dfrac{1}{{{{\sin }^2}x}} = \dfrac{4}{3}$

Taking reciprocal on both the sides.

$$\Rightarrow {\sin ^2}x = \dfrac{3}{4} \\\ \Rightarrow \sin x = \pm \sqrt {\dfrac{3}{4}} \\\ \Rightarrow \sin x = \pm \dfrac{{\sqrt 3 }}{{\sqrt 4 }} \\\ \Rightarrow \sin x = \pm \dfrac{{\sqrt 3 }}{2}' \\\ \Rightarrow x = {\sin ^{ - 1}}\left( { \pm \dfrac{{\sqrt 3 }}{2}} \right) \\\$$

${\sin ^{ - 1}}\left( { \pm \dfrac{{\sqrt 3 }}{2}} \right)$= An angle whose cosine is equal to $\pm \dfrac{{\sqrt 3 }}{2}$.

Hence, $x = \pm \dfrac{\pi }{3} + 2n\pi , \pm \dfrac{{2\pi }}{3} + 2n\pi$where$n \in Z$,$Z$represents the set of all integers.

Therefore, the solution to expression$3{\csc ^2}x = 4$ is$x = \pm \dfrac{\pi }{3} + 2n\pi , \pm \dfrac{{2\pi }}{3} + 2n\pi$ where$n \in Z$,$Z$represents the set of all integers.

Additional Information:
1. Periodic Function= A function $f(x)$ is said to be a periodic function if there exists a real number T > 0 such that $f(x + T) = f(x)$ for all x.

If T is the smallest positive real number such that $f(x + T) = f(x)$ for all x, then T is called the fundamental period of $f(x)$ .

Since $\sin \,(2n\pi + \theta ) = \sin \theta$ for all values of $\theta$ and n$\in$N.
2. Even Function – A function $f(x)$ is said to be an even function ,if $f( - x) = f(x)$for all x in its domain.

Odd Function – A function $f(x)$ is said to be an even function ,if $f( - x) = - f(x)$for all x in its domain.

We know that $\sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta ) = - \tan \theta$

Note: 1. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.

2.One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.

3. $Z$represents the set of all integers.