Inv, co, arc, arcco, inv co, etc

[mewhoexactlywhat]

Can anyone tell me what the difference is, if any, between inverse _, arc_, co_, and _^-1, when refereing to any of the trigonometric ratios? Also, what would arcco_, and inverse co_ refer to? Thank you.

 

[mathman]

Arc, inverse, and ^-1 are all the same thing. co cannot be characterized in general. Cotan is reciprocal of tan, cosine and sine are related by sum of squares =1, secant and cosecant are reciprocals of cosine and sine respectively.

 

[mewhoexactlywhat]

Thank you!

 

[LeonhardEuler]

arc_ and _^-1 both mean the same thing: the inverse of the function. So if [itex]x=sin(\theta)[/tex], then [itex]\theta=arcsin(x)[/tex], which is the same thing as [itex]\theta=sin^{-1}(x)[/tex]. With reference to "co_":the sine and cosine functions are the same, except the cosine function has a phase shifted by [itex]\frac{\pi} {2}[/tex]. Look at the graphs of sin(x) and cos(x) and it will be clear what this means. Other trigonometric functions are derived from sin(x) and cos(x), and in general the "co_" means that everywhere there is a sin(x) in the definition of "_"(x) there is a cos(x) in the definition of "co_"(x) and everywhere there is a cos(x) in "_"(x), there is a sin(x) in "co_"(x). "arcco_", "co_^-1", and "inverse co_" would all just refer to the inverse of the function "co_".

 

[HallsofIvy]

Actually, "co" can be characterized in general- at least for the trig functions.

If θ is an angle in a right triangle, then the angle opposite it is its complement. cosine, cotangent, and cosecant are the sine, tangent, and secant of the complementary angle.