The inverse sine, inverse cosine, and inverse tangent...

The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. (a) The function $\sin ^{-1}$ has domain _____________ and range ______________. (b) The function cos $^{-1}$ has domain _____________ and range _______________. (c) The function tan $^{-1}$ has domain ______________ and range _______________.

The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges.
(a) The function $\sin ^{-1}$ has domain _____________ and range ______________.
(b) The function cos $^{-1}$ has domain _____________ and range _______________.
(c) The function tan $^{-1}$ has domain ______________ and range _______________.

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Key Concepts

Domain and Range in Inverse Functions

The domain of an inverse trigonometric function is determined by the range of its corresponding trigonometric function, while its range is a specifically chosen interval (the principal branch) that ensures the function is single-valued. This restriction is essential to avoid ambiguity in the output values.

Principal Branches

The principal branch of an inverse trigonometric function is the selected interval where the original trigonometric function is bijective. This selection guarantees that every output, that is every angle, is unique, making it possible to systematically solve equations involving these functions.

Inverse Trigonometric Functions

Inverse trigonometric functions return the angle associated with a given trigonometric value. They are defined by restricting the domain of the original trigonometric functions so that they become one-to-one, which is necessary for the existence of an inverse function.

Transcript

0:00 All right.

00:01 So in this problem here, we're trying to find the domain and range of these three different inverse functions, sine, cosine, and tangent inverse.

00:10 So let's start with the domain of sine inverse.

00:13 So basically, what we're looking at here is that when we have, i've drawn the graphs of the sign regular function.

00:19 And we can know that the range of sign regular is negative one, like right here, up to one right here.

00:26 And now for an inverse function, the domain of the inverse function, the domain of the inverse function, function is the same as the range of the original function.

00:33 So now we can just say that the domain is negative one comma one, because that is the range of normal sign.

00:41 As for the range of the inverse sign function, we now need to look at the domain of regular sign.

00:46 And we need to restrict the domain because this did not pass the horizontal line test.

00:51 So we're going to restrict the domain so that we can have all values in the range.

00:55 For sign, this means that we're going to need to draw a line here and draw a line here.

01:00 That the domain is restricted from negative pi over two to pi over two.

01:05 So that is going to be our range for inverse nine, negative pi over two to pi over two.

01:18 Okay...

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