Question

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

Answer 1

`tan(arcsin(1/3)) = sqrt(2)/4`

Explanation:

Consider a right angled triangle with sides `1`, `2sqrt(2)` and `3`

We can tell that it is right angled since:

`1^2+(2sqrt(2))^2 = 1+8 = 9 = 3^2`

Denote the smallest internal angle by `theta`.

Then:

`sin(theta) = "opposite"/"hypotenuse" = 1/3`

`tan(theta) = "opposite"/"adjacent" = 1/(2sqrt(2)) = sqrt(2)/4`