Question

How do you solve $2\sin x + \csc x = 0$ in the interval $0$ to $2\pi?$

Answer 1

Hint: In this question, we are going to solve the given equation for the given interval.
First we are going to write the cosecant $x$ by the reciprocal identity and then multiplying by $\sin x$ on both sides.
And then simplifying the equation we get the result and then we can get the solution from the given interval.
Hence, we get the required solution from the given interval.

Formula used: The reciprocal identity is written as
$\cos ecx = \dfrac{1}{{\sin x}}$

Complete step-by-step solution:
In this question, we are going to solve the given equation by the given interval.
First write the given equation and mark it as $\left( 1 \right)$
$2\sin x + \csc x = 0……….....\left( 1 \right)$
Now by applying the reciprocal identity to the cosecant $x$ in equation $\left( 1 \right)$ we get,
$ \Rightarrow 2\sin x + \dfrac{1}{{\sin x}} = 0$
Multiplying $\sin x$ on both sides of the equation we get,
$ \Rightarrow 2{\sin ^2}x + \dfrac{{\sin x}}{{\sin x}} = 0$
On cancel the term and we get
$ \Rightarrow 2{\sin ^2}x + 1 = 0$
On rewriting the term and we get
\[ \Rightarrow 2{\sin ^2}x = - 1\]
Let us divide the term and we get,
\[ \Rightarrow {\sin ^2}x = \dfrac{{ - 1}}{2}\]
But as ${\sin ^2}x$ cannot be negative
Hence we do not have any solution to $2\sin x + \csc x = 0$
The solution can also be checked from the graph of $2\sin x + \csc x = 0$, which never has a value zero.
Hence the given equation has no solution.

Note: Solving the trigonometric equation is a tricky work that often leads to errors and mistakes. Therefore, answers should be carefully checked. After solving, you can check the answers by using a graph.
The unit circle or trigonometric circle as it is also known is useful to know because it let us easily calculate the cosine, sine, and tangent of any angle between $0$ and $360$ degrees.
Trigonometry is also helpful to measure the height of the mountain, to find the distance of long rivers, etc. its applications are in various fields like oceanography, astronomy, navigation, electronics, physical sciences etc.