Question

Find the exact value of sin(arctan(2)). For full credit, explain your reasoning.

Answer 1

there is a picture of an angle \(\theta\) with \(\tan(\theta)=2\)

Answer 2

in other words \(\theta=\arctan(2)\) and you want \(\sin(\theta)\)
all that is missing is the hypotenuse

Answer 3

which you do in your head via pythatoras
\[h=\sqrt{1^2+2^2}=\sqrt{5}\]

Answer 4

now it should be easy to find \(\sin(\theta)=\sin(\arctan(2))\) it is opposite over hypotenuse

Answer 5

How do you know the adjacent side is 1?

Answer 6

i made it up
\[\tan(\theta)=2\] so the ration of the "opposite" side to the "adjacent" sides is 2

Answer 7

it is easiest to make the opposite side 2 and the adjacent side 1, but i could have made the opposite side 200 and the adjacent side 100
or opposite side 12 and the adjacent side 6

Answer 8

all the ratios would work out the same, just easiest to make it 2 and 1