Question

How does one find the general solution for $\tan x - 3\cot x = 0$ ?

Answer 1

Hint: For the given equation firstly convert the $\cot $ in terms of $\tan $. Then simplify by taking the square root of both sides. Lastly used trigonometric identities and general solution formula for,
$\tan \theta = \tan \alpha , \\
\Rightarrow \theta = \alpha + k\pi $
Where,$k$ is any integer value.

Complete step by step answer:
We have the following equation ,
$\tan x - 3\cot x = 0$ ,
Now by using the trigonometric property i.e., $\cot x = \dfrac{1}{{\tan x}}$ ,
Convert the $\cot $ in terms of $\tan $.
We get ,
$\Rightarrow \tan x - \dfrac{3}{{\tan x}} = 0 \\
\Rightarrow {\tan ^2}x - 3 = 0 $
Where $\tan x \ne 0$ ,
After simplifying ,
$\Rightarrow {\tan ^2}x = 3 \\
\Rightarrow \tan x = \pm \sqrt 3 $
Now we have two values , one with positive sign and another with negative sign.
For $\tan x = \sqrt 3 $ ,
$ \Rightarrow x = \dfrac{\pi }{3}$
For this the general solution is
$ \Rightarrow x = \dfrac{\pi }{3} + k\pi $
Where, $k$is any integer value .
And for $\tan x = - \sqrt 3 $ ,
$ \Rightarrow x = \dfrac{{2\pi }}{3}$
For this the general solution is
$ \therefore x = \dfrac{{2\pi }}{3} + k\pi $
Where ,$k$ is any integer value.

Additional Information:
Trigonometric equations can have two types of solutions one is principal solution and other is general solution. Principal solution is the one where the equation involves a variable $0 \leqslant x \leqslant 2\pi $ . General solution is the one which involves an integer, say $k$ and provides all the solutions of the trigonometric equation.

Note: You must know that sin, cosine, and tangent are considered as major trigonometric functions, hence we can derive the solutions for the equations comprising these trigonometric functions or ratios. We can also derive the solutions for the other three trigonometric functions such as secant, cosecant, and cotangent with the help of the solutions which are already derived.