# How do you find the exact value of tan(arcsin(1/3))?

Feb 9, 2017

$\tan \left(\arcsin \left(\frac{1}{3}\right)\right) = \frac{\sqrt{2}}{4}$

#### Explanation:

Consider a right angled triangle with sides $1$, $2 \sqrt{2}$ and $3$

We can tell that it is right angled since:

${1}^{2} + {\left(2 \sqrt{2}\right)}^{2} = 1 + 8 = 9 = {3}^{2}$

Denote the smallest internal angle by $\theta$.

Then:

$\sin \left(\theta\right) = \text{opposite"/"hypotenuse} = \frac{1}{3}$

$\tan \left(\theta\right) = \text{opposite"/"adjacent} = \frac{1}{2 \sqrt{2}} = \frac{\sqrt{2}}{4}$