1. ## Inverse Trigonometric Functions

Establishing an Identity Involving Inverse Trigonometric Functions.

(1) Show that tan[arcsin(v)] = v/(sqrt{1 - v^2})

(2) Show that sin[arccos(v)] = sqrt{1 - v^2}

2. #1: Recall that $\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}$

So: $\displaystyle \tan \left(\arcsin v\right) = \frac{\sin \left(\arcsin v\right)}{\cos \left(\arcsin v\right)} = \frac{v}{\sqrt{1 - \big[\sin \left(\arcsin v\right)\big]^2}} = ...$

#2: Similar to what was done in the denominator of #1, recall that $\displaystyle \cos^2 x = 1 - \sin^2x$

So: $\displaystyle \sin \left(\arccos v\right) = \sqrt{1 - \big[\cos \left(\arccos v\right)\big]^2}$

3. Originally Posted by magentarita
Establishing an Identity Involving Inverse Trigonometric Functions.

(1) Show that tan[arcsin(v)] = v/(sqrt{1 - v^2})

(2) Show that sin[arccos(v)] = sqrt{1 - v^2}
#1: If we were to plot a triangle that would represent $\displaystyle \sin^{-1}v$, the leg opposite the angle would have a value of v, and the hypotenuse would have a value of one.

To solve for the side adjacent to the angle, apply pythagoras' theorem:

$\displaystyle v^2+b^2=1\implies b=\sqrt{1-v^2}$

Now, since $\displaystyle \tan\vartheta=\frac{opp}{adj}$, and $\displaystyle opp=v~and~adj=\sqrt{1-v^2}$ we see then that $\displaystyle \tan\vartheta=\tan\left(\sin^{-1}v\right)=\frac{v}{\sqrt{1-v^2}}$

$\displaystyle \mathbb{Q.E.D.}$

#2: If we were to plot a triangle that would represent $\displaystyle \cos^{-1}v$, the leg adjacent to the angle would have a value of v, and the hypotenuse would have a value of one.

To solve for the side opposite to the angle, apply pythagoras' theorem:

$\displaystyle v^2+b^2=1\implies b=\sqrt{1-v^2}$

Now, since $\displaystyle \sin\vartheta=\frac{opp}{hyp}$, and $\displaystyle opp=\sqrt{1-v^2}~and~hyp=1$ we see then that $\displaystyle \sin\vartheta=\sin\left(\cos^{-1}v\right)=\sqrt{1-v^2}$

$\displaystyle \mathbb{Q.E.D.}$

Does this make sense??

--Chris

4. ## Thanks

Thank you both. Of course, Chris' method is much easier to follow.

5. Hello, magentarita!

Another approach . . .

Establishing an identity involving inverse trigonometric functions.

(1) Show that: .$\displaystyle \tan[\arcsin(v)] \:= \:\frac{v}{\sqrt{1 - v^2}}$
Recall that an inverse trig expression is an angle.

We have: .$\displaystyle \tan\underbrace{[\arcsin(v)]}_{\text{some angle }\theta}$

That is: .$\displaystyle \theta$ is an angle whose sine is $\displaystyle v.$

We have: .$\displaystyle \sin\theta \:=\:\frac{v}{1} \:=\:\frac{opp}{hyp}$

$\displaystyle \theta$ is in a right triangle with: .$\displaystyle opp = v,\;hyp = 1$
. . and we want: .$\displaystyle \tan\theta$
Code:
                      *
1    *   |
*       | v
* θ         |
* - - - - - - - *
____
√1-vē
Using Pythagorus, we find that .$\displaystyle adj = \sqrt{1-v^2}$

Therefore: .$\displaystyle \tan\theta \:=\:\frac{opp}{adj} \:=\:\frac{v}{\sqrt{1-v^2}}$

Ahh, Chris beat me to it . . .
.

6. ## soroban

Originally Posted by Soroban
Hello, magentarita!

Another approach . . .

Recall that an inverse trig expression is an angle.

We have: .$\displaystyle \tan\underbrace{[\arcsin(v)]}_{\text{some angle }\theta}$

That is: .$\displaystyle \theta$ is an angle whose sine is $\displaystyle v.$

We have: .$\displaystyle \sin\theta \:=\:\frac{v}{1} \:=\:\frac{opp}{hyp}$

$\displaystyle \theta$ is in a right triangle with: .$\displaystyle opp = v,\;hyp = 1$
. . and we want: .$\displaystyle \tan\theta$
Code:
                      *
1    *   |
*       | v
* θ         |
* - - - - - - - *
____
√1-vē
Using Pythagorus, we find that .$\displaystyle adj = \sqrt{1-v^2}$

Therefore: .$\displaystyle \tan\theta \:=\:\frac{opp}{adj} \:=\:\frac{v}{\sqrt{1-v^2}}$

Ahh, Chris beat me to it . . .
.
Thank you Soroban.