Question

Find the general solution for tanx = $\sqrt{3}$

Answer 1

Hint: First we will write that for what value of tan of the angle we get $\sqrt{3}$ , and then we will use the general solution of tan to find all the possible solutions, and we can see that there will be infinitely many solutions of x for which it gives tanx = $\sqrt{3}$ .
Complete step-by-step answer:
Let’s start solving the question.
Let’s first find the value of angle for which we get $\sqrt{3}$.
Now we need to find that at which quadrant tan is positive,
We know that tan is positive in ${3}^{rd}$ and ${1}^{st}$ quadrant, so $\dfrac{\pi }{3}$ and $\pi +\dfrac{\pi }{3}$ both are the correct value,
Here, we will take $\dfrac{\pi }{3}$.
Now we know that $\tan \dfrac{\pi }{3}=\sqrt{3}$
Hence, we get $\tan x=\tan \dfrac{\pi }{3}$
Now we will use the formula for general solution of tan,
Now, if we have $\tan \theta =\tan \alpha$ then the general solution is:
$\theta =n\pi +\alpha$
Now using the above formula for $\tan x=\tan \dfrac{\pi }{3}$ we get,
$x=n\pi +\dfrac{\pi }{3}$
Here n = integer.
Hence, from this we can see that we will get infinitely many solutions for x as we change the value of n.

Note: The formula for finding the general solution of tan is very important and must be kept in mind.
In the above solution we have taken the value of $\alpha$ we have taken was$\dfrac{\pi }{3}$ , but one can also take the value of $\alpha$ as $\dfrac{4\pi }{3}$ , as it lies in the ${3}^{rd}$ quadrant and gives positive value for tan. And then one can use the same formula for a general solution and replace the value of $\alpha$ with $\dfrac{4\pi }{3}$ to get the answer, which is also correct.