trigonometry

[t.man]

some identities.
it is true that sin(sin^-1 (x) = x
i.e if sin(sin^-1 (x)) = y
then sin^-1(y) = sin^-1(x) => x = y;

but what cos(sin^-1(x)) equals?
i.e if cos^-1(x) = sin^-1(y). what can one say about x and y?

 

[dknguyen]

some identities.
it is true that sin(sin^-1 (x) = x
i.e if sin(sin^-1 (x)) = y
then sin^-1(y) = sin^-1(x) => x = y;

but what cos(sin^-1(x)) equals?
i.e if cos^-1(x) = sin^-1(y). what can one say about x and y?

Just graph it out. I don't have any paper on hand but they might not intersect. Not everything has to have a nice neat analytical answer.

 

[skyhawk]

but what cos(sin^-1(x)) equals?
i.e if cos^-1(x) = sin^-1(y). what can one say about x and y?

x^2 + y^2 = 1.

 

[t.man]

x^2 + y^2 = 1.

which means i can safely say:

cos(sin^-1(x))= square root of (1 - x^2)

 

[skyhawk]

which means i can safely say:

cos(sin^-1(x))= square root of (1 - x^2)

It depends on your purpose. If you are restricting your answer to the usual principal values where arcsin is restricted to -pi/2 to pi/2 and arccos is restricted to 0 to pi then the answer is yes. However, if you are solving a (physical) problem where the answer can lie in any quadrant then you need to include the negative possibility also.

 

[MrAl]

some identities.
it is true that sin(sin^-1 (x) = x
i.e if sin(sin^-1 (x)) = y
then sin^-1(y) = sin^-1(x) => x = y;

Hi,

What exactly are you asking here?
If sin(asin(x))=x then why would you want to state that sin(asin(x))=y , ie what are you trying to prove?

Yes, cos(asin(x))=sqrt(1-x^2).

Complex numbers allow us to go further with this too, for example when x=2.