Inverse Function Homework: Simplifying Sin^-1(2Sin^-1(0.8))

[Karol]

Homework Statement

Simplify:
$$\sin^{-1}(2\sin^{-1}0.8)$$

Homework Equations

Inverse sine: $y=\sin^{-1}(x)~\rightarrow~\sin(y)=x$
$$\sin^2(x)+\cos^2(x)=1$$

The Attempt at a Solution

The inner parenthesis: $\sin y=0.8$ . In the drawing it's alpha's sine.

Now i double the α and the question wants the high edge in the drawing. how to find it?

 

[SammyS]

Homework Statement

Simplify:
$$\sin^{-1}(2\sin^{-1}0.8)$$

Homework Equations

Inverse sine: $y=\sin^{-1}(x)~\rightarrow~\sin(y)=x$
$$\sin^2(x)+\cos^2(x)=1$$

The Attempt at a Solution

The inner parenthesis: $\sin y=0.8$ . In the drawing it's alpha's sine.
View attachment 109084 Now i double the α and the question wants the high edge in the drawing. how to find it?

That problem seems very strange to me.

It would be much more expected to be asked to simplify something like:

$\sin\left(2 \sin^{-1} (0.8)\right)$

 

[member 587159]

Homework Statement

Simplify:
$$\sin^{-1}(2\sin^{-1}0.8)$$

Homework Equations

Inverse sine: $y=\sin^{-1}(x)~\rightarrow~\sin(y)=x$
$$\sin^2(x)+\cos^2(x)=1$$

The Attempt at a Solution

The inner parenthesis: $\sin y=0.8$ . In the drawing it's alpha's sine.
View attachment 109084 Now i double the α and the question wants the high edge in the drawing. how to find it?

Are you sure that's the correct question? It seems undefined to me.

 

[haruspex]

That problem seems very strange to me.

It would be much more expected to be asked to simplify something like:

$\sin\left(2 \sin^{-1} (0.8)\right)$

Looking at the diagram, that is how Karol interpreted it.
@Karol, what formulae do you know for sin(2α) or sin(α+β)?

 

[Karol]

$$\sin(2\alpha)=2\sin(\alpha)\cos(\alpha)$$
$$\sin^2(\alpha)+\cos^2(\alpha)=1~\rightarrow~\cos(\alpha)=0.6$$
$$\sin(2\alpha)=2\cdot 0.8 \cdot 0.6$$

 

[SammyS]

$$\sin(2\alpha)=2\sin(\alpha)\cos(\alpha)$$
$$\sin^2(\alpha)+\cos^2(\alpha)=1~\rightarrow~\cos(\alpha)=0.6$$
$$\sin(2\alpha)=2\cdot 0.8 \cdot 0.6$$

That looks fine, if you're trying to find $\ \sin\left(2 \sin^{-1} (0.8)\right) \, .$

 

[member 587159]

$$\sin(2\alpha)=2\sin(\alpha)\cos(\alpha)$$
$$\sin^2(\alpha)+\cos^2(\alpha)=1~\rightarrow~\cos(\alpha)=0.6$$
$$\sin(2\alpha)=2\cdot 0.8 \cdot 0.6$$

Also, if you want to type an implication '$\Rightarrow$', write 'Rightarrow' in Latex instead of 'rightarrow'.

 

[Karol]

Thanks everybody, you are great!