# How to find $\tan{\theta}$ when $\theta=\arctan⁡{(8/3)}$

Basically I'm trying to find the exact value of $\tan{\theta}$ when $\theta = \arctan{(8/3)}$.

I'm not exactly sure where to start. I know that $\arctan$ is the inverse of $\tan$, but I can't really figure out how to do the inverse of this one.

I also have to find $\sin{\theta}$, but I feel that will be easier once I find $\tan{\theta}$.

• $\tan(\arctan x)=x$, of course… Jun 13, 2016 at 1:09
• To get $\ \sin \theta \$ , construct a right triangle with an angle $\ \theta \$ having an "opposite side" of $\ 8 \$ and an "adjacent side" of $\ 3 \$ : find the hypotenuse and you can get any of the trig values for $\ \theta \$ that you may want. Jun 13, 2016 at 1:11
• Do tell your quadrant, or all will come to grief. Jun 13, 2016 at 1:14

$$\tan(\theta) = \tan(\arctan{(8/3)} ) = 8/3.$$
$$\sin(\theta) = \pm\frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}} = \pm \frac{8}{\sqrt{3^3+8^2}} = \pm \frac{8}{\sqrt{73}}.$$
• dummies.com/how-to/content/… . For $\cos(\theta)$, you will have $\pm \sqrt{1-\sin^2(\theta)} = \pm \frac{3}{\sqrt{73}}$. Jun 13, 2016 at 2:24