Thread: Inverse Trigonometric Fnnction and when they cancel what their inverse does

1. Inverse Trigonometric Fnnction and when they cancel what their inverse does

When do the inverse trigonometric functions cancel each other? I mean that while sin(arcsin(2/3))=2/3, you can't just do that here: arcsin(sin(5π/13)). when do they cancel each other and when you need to be careful? Thanks.

In my homework I have examples that show that sin(arcsin(2/3))=2/3. So it is clear that the functions cancel each other. But in another example: arcsin(sin(5π/13π). Here, it shows a little work before they cancel what sin does: arcsin(sin(5π/13)=arcsin(sin(π−5π/13))=2π/7.

2. Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

Originally Posted by IloveIl
When do the inverse trigonometric functions cancel each other? I mean that while sin(arcsin(2/3))=2/3, you can't just do that here: arcsin(sin(5π/13)). when do they cancel each other and when you need to be careful? Thanks.

In my homework I have examples that show that sin(arcsin(2/3))=2/3. So it is clear that the functions cancel each other. But in another example: arcsin(sin(5π/13π). Here, it shows a little work before they cancel what sin does: arcsin(sin(5π/13)=arcsin(sin(π−5π/13))=2π/7.

Have a look at this table.

Try to explain to yourself. Lookup the domain & range of the $\arcsin$ function.

3. Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

but 5π/13 is outside the domain of arcsine.
true, however the expression is $\arcsin\left[\sin\left(\dfrac{5\pi}{13}\right)\right]$, not $\arcsin\left(\dfrac{5\pi}{13}\right)$

$0 < \sin\left(\dfrac{5\pi}{13}\right) < 1$ is in the domain of the arcsine function ...

4. Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

Accidental double post.

5. Re: Inverse Trigonometric Fnnction and when they cancel what their inverse does

First, the trig functions, because they are not "one to one" do not have true "inverses". In order to have inverses, we have to restrict the domains. For example, we can take x from $\displaystyle -\frac{\pi}{2}$ to $\displaystyle \frac{\pi}{2}$ without values repeating. So arcsin(y) always lies between $\displaystyle -\frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$.

But your example, $\displaystyle arcsin(sin(\frac{5\pi}{13}))$, does not illustrate that. $\displaystyle \frac{5\pi}{13}$ is less than $\displaystyle \frac{\pi}{2}$ so $\displaystyle arcsin(sin(\frac{5\pi}{13}))= \frac{5\pi}{13}$! Using a calculator, $\displaystyle \frac{5\pi}{13}= 1.2083$, $\displaystyle sin(1.2083)= 0.9350$, and $\displaystyle arcsin(0.9350)= 1.2083$.