Question

Find the general solution of the trigonometric equation given by, $\sin x = \tan x$

Answer 1

To find the general solution we express the function in terms of sin and cos trigonometric functions. We then reduce the equation into a single function and find the solution to it.

Complete step-by-step answer :
Given that,
$\sin x = \tan x$ ……… (i)
We know that,
$\tan x = \dfrac{{\sin x}}{{\cos x}}$,
Put this in equation (i),
$\Rightarrow \sin x = \dfrac{{\sin x}}{{\cos x}}$
$\Rightarrow \sin x\cos x = \sin x$
This can be written as:
$\Rightarrow \cos x = \dfrac{{\sin x}}{{\sin x}}$
$\Rightarrow \cos x = 1$
Now, we have to find the value of x, for which $\cos x = 1$
We know that,
$\cos 0 = 1$
Therefore,
$\cos x = \cos 0$
We know that,
$\cos x = \cos y$, implies $x = 2n\pi \pm y$, where $n \in Z$ [Z – set of integers]
Therefore,
$\cos x = \cos 0$
Implies,
$x = 2n\pi \pm 0$ or,
$x = 2n\pi$ where $n \in Z$ [Z is a set of integers]
Hence, the general solution of $\sin x = \tan x$ is $x = 2n\pi$

Note : In order to solve this type of problems the key is to know the values of angles (x) for some frequent/general values of the function Cos x. We have to remember that for any real numbers x and y, $\cos x = \cos y$, implies $x = 2n\pi \pm y$, where $n \in Z$ and Z is a set of integers and thus, we get the general solution.