Does sin squared x equal (sin x)^2 and does sin to the -1 x equal (sin x)^-1?

Answer 1

yes and no. In that order.

Answer 2

The answer is yes to the first and no to the second.

sin^2(x) is the same as (sin(x))^2.

sin^-1(x) is NOT the same as (sin(x))^-1.

sin^-1(x) denotes the arcsin which means that if y=sin(x) then arcsin(y) = x. So arcsin is the "inverse" of the sin function.

(sin(x))^-1 denotes the inverse as well BUT it denotes the multiplicatiev inverse. Which means that
(sin(x))^-1 = 1/sin(x) = csc(x).
This is just like how we do it with numbers.
2^-1 = 1/2
3^-1 = 1/3
100^-1 = 1/100

So yes, there is a difference. Just to remind you that this same convention is true for ALL trignometric functions.

cos^-1(x) = arccos(x), called inverse cosine
tan^-1(x) = arctan (x), inverse tangent
cot^-1(x) = arccot(x), inverse cotangent
sec^-1(x) = arcsec(x), inverse secant
csc^-1(x) = arccsc(x), inverse cosecant

(cos(x))^-1 = 1/cos(x) = sec(x)
(tan(x))^-1 = 1/tan(x) = cot(x)
and then their opposites....


Here is the REAL mind bender, some books/professors even distinguish between Arcsin and arcsin. The one with the capital A, represents the cosine inverse FUNCTION and the one with the lower case a, represents the inverse relation.

Meaning that y=Arcsin(x) is a FUNCTION defined as
y=Arcsin(x)=arcsin(x) for -1
y=arcsin(x) is simply sin(y)=x

This is because, if you remember that if I simply take the inverse of the sine function, the inverse is not a function (because it will fail the vertical line test) so we restrict the domain to MAKE it a function.

And again, this is true for all trig functions.

Answer 3

sin squared x = (sin x)^2

sin to the -1 x is not (sin x)^-1

sin to the -1 x stands for arcsin(x)

that is you are looking for the the angle y such that x = sin(y)

Answer 4

sine squared x is the same as (sin x)^2

But sine to the -1 x is called inverse sine of x or arcsin x, which is basically saying this:

If:
sin a = x
Then:
sin ^(-1) x = arcsin x = a

Good luck! =)

Answer 5

I had the same question when I was in college. The answer is yes to all.

Answer 6

yes to both

Answer 7

the answer is blue