Using the identity $\tan\theta = \tan(\theta-180^\circ)$ to find values of $\theta$ such that $\tan\theta=\tan 20^\circ$

Question

This is question 1c of a list of related items.

State the value(s) of $\theta$ in the range $0^\circ$ to $360^\circ$ so that the following is true: $$\tan\theta = \tan 20^\circ$$

Here is the answer (from the list of answers):

$$\theta = 20^\circ;\quad \theta=180^\circ+20^\circ=200^\circ$$

I am using the trig identity for tan, the one where $$\tan\theta = \tan(\theta-180^\circ)$$ If $\theta = 20^\circ$ for the question, then $\tan(\theta-180^\circ)$ is $\tan(20^\circ-180^\circ)$, which is $\tan(-160^\circ)$, which is taking me to a completely different direction than the solution.

I would appreciate it if someone could explain the steps of using this trig identity to determine which other angles have the same tan ratio as $20^\circ$.

Answer 1

Hint:period of tangent function is $180^\circ, \tan 20^\circ=\tan(180+20)^\circ, \tan(20-180)^\circ =\tan(-160)^\circ, 200^\circ$ is anticlockwise rotation, $-160^\circ$ is clockwise rotation.

Answer 2

Since $\tan \theta = \tan (n\pi + \theta)$, where $n$ is any integer and $\theta=\dfrac{\pi}{9}$ in our case. So, we have $0 \leq n\pi+\dfrac{\pi}{9} \leq 2\pi \implies 0 \leq (9n+1)\pi \leq 18\pi$. Now you can clearly see that only $n=0$ and $n=1$ satisfy this bound, hence the only answers are $\theta=\dfrac{\pi}{9},\dfrac{10\pi}{9}$.

Answer 3

$$\theta = 20^\circ;\quad \theta=20^\circ\pm n\cdot 180^\circ;$$

Other angles are

$$ =20^\circ,200^\circ,380^\circ,460^\circ $$

We can add or subtract $ n\cdot \pi$ ( $n$ positive or negative integer) to $ 20^\circ $ getting the tip of radius vector from first to third quadrant. The radius vector can be rotated indefinitely around the origin adding $ 180^\circ$.

Answer 4

You suggest we use that $$ \tan \theta = \tan(\theta - 180^\circ). $$ Equivalently, if we set $\phi = \theta - 180^\circ$, the above equation reads $$ \tan(\phi+180^\circ) = \tan\phi. $$ To summarize: the tangent function is $180^\circ$-periodic: it repeats every $180^\circ$. So the possible answers are $\dots,-340^\circ,-160^\circ, 20^\circ, 200^\circ, 380^\circ, \dots$

Narasimham's answer shows how to express this in a single equation with $n$ varying, ABCD does the same in the language of radians ($180^\circ = \pi$ radians).