difference between (1/sin) and sin^(-1)?

Question

they are not the same thing. even if you put it in the calculator (TI-83) it gives you different answers. sin^(-1)=1.5707 1/sin(1)=1.18839

sorry, i miss typed some information, i set x=1 so sin^-1(1)=1.5707 and 1/sin(1)=1.18839

Answer 1

It is extremely important to realize that 1/sin "one over sine" and sin^(-1) "sine inverse" are NOT the same.

Even though sin^2(x) is a conventional way of saying (sin(x))^2, that does NOT mean sin^(-1)(x) is the same as (sin(x))^(-1).

sin^(-1)(x) is the function such that sin(sin^(-1)(x)) = x. That is, it's the FUNCTIONAL inverse of sin(x).

1/sin(x) is the multiplicative inverse.

Do this on your calculator:

sin^(-1)( 0.5 )

And then do this on your calculator:

1/(sin [0.5])

And you'll find out they are not the same.

To obtain sin^(-1) on your calculator, usually you'd have to use the "2nd function" button.

Suppose that

sin^-1(x)=y. Then by definition,

siny = x
sin^-1[1/2] = 30 degrees,
because sin(30) = 1/2.

Now, take f(x) = 1/sin(x)

Let x = 1/2.

Then f(1/2) = 1/sin(1/2), and we don't know the sine of 1/2 because it's not on our known list of values.

Answer 2

Raj's, Puggy's, and Blue Moon's answers are all the right ones. I just want to throw in my 2 cents.

This is a classic example of mathematicians being lazy and imprecise. This happens even with math guys, unfortunately.

The problem is that they are using an exponent to mean two different things.

To be consistent, no one should ever write:

sin² x

When what they really mean is

(sin x)²

Because "sin(x)" is the function, and then you're squaring the result; you're not actually finding the sin² of x.

But people get lazy and hate to write parentheses all the time, so they abbreviate it as sin²(x).

Then, there's a problem comes when they use the exponent -1 to mean "the inverse function".

In my opinion, this is a more legitimate use of the exponent, because you are really doing something to the function itself, rather than to the output of the function.

(Since a function is like a machine that takes an input and produces an output, the "inverse" of a function is like running the machine backwards, giving it an output, and getting out an input. For example, if f(3) = 7, then f(^-1)(7) = 3 (unless f is not 1-to-1, but let's ignore that).)

In any case, whenever a function is raised to the -1 power, it is assumed that what is meant is the "inverse" of that function, rather than "calculate the answer, and flip it".

So the sin^(-1)(x) means, "the angle whose sine is equal to x".

Just to avoid this kind of confusion, you can also write sin^(-1)x as arcsin x.

Answer 3

Note the is a difference between sin^-1(x) and (sin(x))^-1.

Answer 4

With the latter, we often write arcsin. (likewise arccos, arctan, etc.)

1/sin(x) is taking 1 and dividing it by the sin of the angle x, where x is an angle and we receive the y-component of our unit circle (between -1 and 1).

arcsin(x) is finding the angle who's sin value equals x, where x is the y-component of our unit circle (btwn. -1 and 1) and we receive the corresponding angles.

Answer 5

1/sin is reciprocal of sin
sin^-1 is the inverse of sin

Answer 6

They are one in the same.

Do this on your calculator:

1/5

and

5^-1

You will find out they are the same number, 0.2 Raising a number, even sin, to the negative one power just means taking the inverse of it.

Answer 7

suppose
sin^-1[x]=y
then by definition
siny=x
sin^-1[1/2]=?
30Deg
because sin30=1/2
sinx and 1/sinx
is the same as
4 and 1/4

Answer 8

it's the same thing.
you must be wrong when you put the variables, just make sure that you put the parenthesis correctly.