# Thread: Calculating the Exact Value of Trig Functions

1. ## Calculating the Exact Value of Trig Functions

Hi Guys,

I have another question:calculate the exact value of the following expression:

$\displaystyle tan(arcsin(\frac{2}{3}))$
I don't know how to calculate the exact value of $\displaystyle arcsin(\frac{2}{3})$

Can anyone help?

2. Originally Posted by spycrab
Hi Guys,

I have another question:calculate the exact value of the following expression:

$\displaystyle tan(arcsin(\frac{2}{3}))$
I don't know how to calculate the exact value of $\displaystyle arcsin(\frac{2}{3})$

Can anyone help?
1. Draw a sketch. (see attachment)

2. Use proportions:

$\displaystyle \dfrac{\frac23}1 = \dfrac{y}{\sqrt{1+y^2}}$

3. Since $\displaystyle y = \tan\left(\arcsin\left(\frac23}\right) \right)$ solve the equation at 2. for y.

4. I've got $\displaystyle y = \tan\left(\arcsin\left(\frac23}\right) \right) = \dfrac25 \sqrt{5}$

3. thanks!

4. Hello, spycrab!

Calculate the exact value of: .$\displaystyle \tan\left[\arcsin (\frac{2}{3}\right]$

Let $\displaystyle \theta = \arcsin(\frac{2}{3})\right$

Then: .$\displaystyle \sin\theta \:=\:\frac{2}{3} \:=\:\frac{opp}{hyp}$

That is, $\displaystyle \theta$ is in a right triangle with $\displaystyle opp = 2,\;hyp = 3$

Pythagorus tells us that: .$\displaystyle adj \,=\,\sqrt{5}$

Hence: .$\displaystyle \tan\theta \:=\: \frac{opp}{adj} \:=\:\frac{2}{\sqrt{5}}$

Therefore: .$\displaystyle \tan\left[\arcsin(\frac{2}{3})\right] \;=\;\dfrac{2}{\sqrt{5}}$

5. thanks