# How do you simplify the expression Sin(Arctan 3)?

Feb 20, 2017

$\sin \left(\arctan \left(3\right)\right) = \frac{3}{\sqrt{1 + {3}^{2}}}$

#### Explanation:

$\sin \left(\arctan \left(x\right)\right) = \frac{x}{\sqrt{1 + {x}^{2}}}$ so

$\sin \left(\arctan \left(3\right)\right) = \frac{3}{\sqrt{1 + {3}^{2}}}$

Feb 20, 2017

$\sin \left(\arctan 3\right) = \frac{3}{\sqrt{10}}$

#### Explanation:

Let $\arctan 3 = x$

and therefore $\tan x = 3$

This leads to $\sec x = \sqrt{1 + {\tan}^{2} x} = \sqrt{1 + {3}^{2}} = \sqrt{10}$

and $\cos x = \frac{1}{\sqrt{10}}$

$\therefore$ $\sin \left(\arctan 3\right) = \sin x = \sin \frac{x}{\cos} x \times \cos x$

= $\tan x \times \cos x = 3 \times \frac{1}{\sqrt{10}}$

i.e. $\sin \left(\arctan 3\right) = \frac{3}{\sqrt{10}}$