Both Sin-¹x and Arcsin x are one and the same, and both are in use in Mathematics Textbooks.

To make things simple, if Sin x = 1/2,

it follows that Sin -¹(1/2) = x

or ArcSin(1/2) = x]]>

Your have two types of trigonometric functions:Direct and Inverse

...sine (sin) and cosine(cos) are the basic blocks for building all trigonometric functions

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1)Direct trig. functions

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sin(x)

cos(x)

..now you can build others:

tan(x)=sin(x)/cos(x)

cosec(x)=1/sin(x)

sec(x)=1/cos(x)

Remember the following (it helps not messing up with secant and cosecant until you get used to them) :

[S]in -> [C]osec

[C]os -> [S]ec

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2) Inverse trig. functions

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direct.trig.func. <=> associated inverse func

sin(y)=x <=> y=arcsin(x)

cos(y)=x <=> y=arccos(x)

tan(y)=x <=> y=arctan(x)

sec(y)=x <=> y=arcsec(x)

cosec(y)=x <=> y=arccosec(x)

An inverse function "undoes" what its associated function "does"

examples:

... if sin(A)=0.5 then A=arcsin(0.5)..

... arctan[tan(K)]=K

... arccosec(cosec(T)=0.1 <=> T=0.1

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3) Inverse funcs and calculators

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the functions you see in calculators: sin-¹( cos-¹( tan-¹( etc... are the inverse arc functions.

how to calculate arcsin(0.1)?

In your calc. *should* exist a key or menu with the 'command' SIN-¹(

just enter SIN-¹(0.1) and the result shows up on the display

Same applies to arccos: COS-¹( and arctan:TAN-¹(

->But most calculators don't have SEC/SEC-¹ and COSEC/COSEC-¹...(WHY TE*AS INSTRUME*S????? :@#$#%"!!! >:( so you must use the other to build this ones...

btw..

If you want my personal opinion, I think its wrong naming those functions "something-¹()". If the notation ARC(...) was invented, why the hell calculator manufacturers decided to replace it by (func)-¹? ...It has some logic since the inverse of x is 1/x, or x-¹ .But the concept is a bit different as far as trigonometric functions are concerned so it would be more correct if calcs used the notation arcsin( arctan( arccos(...etc as all math students use.

p.s:I didn't mention hyperbolic trig. functions in order to give you a more "simple" answer. But if you'r curious I can explain them here also

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