How many solutions has the equation $\tan (3x) = 1$?

Question

I have this equation:

$\tan(3x) = 1$

I've come to the following solutions:

• $x_1 = 15$, so the angle is 45º
• $x_2 = 75$, so the angle is 225º

But i've been told that there're up to six posible solutions fo $x$, and I don't think so. I've been watching the unit circle and there're only two possible tangents that has a value of 1.

My teacher insists that there're six solutions, but is it really possible without exceding the 360º?

Answer 1

Just $$3x=45^{\circ}+180^{\circ}k,$$ where $k$ is an integer number.

Now, solve the following system: $$0^{\circ}\leq15^{\circ}+60^{\circ}k\leq360^{\circ}.$$

Answer 2

Note that it isn't the "angle" (input to the tangent function) which is required to be below $360^\circ$, it is $x$ itself.

So the next solution is $x_3=135^\circ$. And so on.