Inverse and Sin Inverse

[NSX]

When someone says "A is inversely proportional to B, it means A=k/B", where k is some proportionality constant.

Or 2 inverse is 1/2, or 0.5, or 2^-1 = 1/2.

Why is that sin inverse, although written as sin^-1 is not 1/sin?
Likewise for cosine and tangent.

[fafalone]

It's properly written arcsine. When you take the sine of a number, it gives you a decimal result. Arcsine takes a decimal result and turns it in to the angle measure.

[NSX]

Originally posted by fafalone
It's properly written arcsine. When you take the sine of a number, it gives you a decimal result. Arcsine takes a decimal result and turns it in to the angle measure.

Ah..so ie. sin(45) = 0.707106781..., this would be sine?
AND

arcsine (0.707106781...) = 45?

[blike]

Yes BUT that is not the only solution, 135 degrees would also be a solution [remember the unit circle], although I believe most calcs only give one solution.

[fafalone]

since arcsin is only defined for -1<x<1, there is only one possible value for arcsin for any number in that range.

[blike]

Ah, that would be why. I suppose its left to the student to calculate other solutions.

[fafalone]

there are no other solutions. multiple solutions for sine, not arcsine.

[blike]

There was a problem set in trig last semester that required us to use the inverse sine function to find an angle, then find the other angle using the angle provided by inverse sin.

[NSX]

Cool. Thanks guys!
So is it called arctangent and arccosine for the tangent and cosine respectively?

[fafalone]

yeah.

as well as arcsecant, arccosecant, arccotangent, etc

[NSX]

So what about functions like cosecant, secant, etc.?

Do they have a inverse like opposites too?

[NSX]

:feedback:

[dave]

Originally posted by NSX
So what about functions like cosecant, secant, etc.?

Do they have a inverse like opposites too?

yeah, but they're fairly useless.

e.g. if you have the equation:

cosec(x) = 2

then it's pretty obvious that this is just the same as

sin(x) = 1/2

and x = arcsin(1/2) = :pi:/6

so it's just another set of functions that don't really have too much use :)

[dave]

oh, also, you can define arccosecant etc in terms of arcsin etc. say you have an equation:

cosec(a) = b
then a = arccsc(b)

but also, 1/sin(a) = b
then a = arcsin(1/b)

but a = arccsc(b), so arcsin(1/b) = arccsc(b)

you can do a similar thing for arcsecant and arccotangent, but it's all fairly useless :)