About this video
The video graphs arc cosine and arc sine by flipping the input and output of known cosine and sine values. Arc cosine runs from (1, 0) to (-1, pi), and arc sine runs from (-1, -pi/2) to (1, pi/2). Both have domain from -1 to 1 and look like one sideways piece of the original wave, restricted so they stay functions.
You'll be able to
- use known cosine and sine values to find points on the graphs of arccos(x) and arcsin(x).
- sketch the graphs of arccos(x) and arcsin(x) and state their domain.
- explain why an inverse trig graph looks like a sideways piece of the original trig graph.
Key words
- arc cosine
- The inverse of cosine: arccos(x) gives the angle whose cosine is x, so arccos(1) = 0 because cos(0) = 1.
- arc sine
- The inverse of sine: arcsin(x) gives the angle whose sine is x, so arcsin(1) = pi/2.
- inverse function
- A function that undoes another by swapping inputs and outputs, which flips the graph's axes.
- domain
- The set of allowed inputs; for arccos and arcsin it is from -1 to 1.
Transcript
- [Instructor] What I wanna do in this video is get a sense of the graphs of arc cosine and arc sine, and arc sine, and to help us there, I set up a little bit of some axes and I've also made a little table of some of the typical values of cosine and sine. And as we already know, arc cosine and arc sine can be viewed as the inverses of cosine and sine.
So for example, if cosine of zero is one, then that means that the arc cosine of one is zero. So I could write arc, let me write that in the same color, so I could write that the arc cosine of one is equal to zero. So let's use that information now to graph what arc cosine might look like.
The arc cosine of one is going to be zero. So if X is one, then we're going to be at zero, right over there. The arc cosine of one half is going to be pi over three. So if we input one half into arc cosine, it's going to output pi over three. Arc cosine of zero is going to be pi over two.
Zero is going to be pi over two. Arc cosine of two pi over, sorry, arc cosine of negative one half is gonna be two pi over three, negative one half gets us to two pi over three. And I'm just finding some select points. But then finally, these are some of the easier ones.
If I take the arc cosine of negative one, I'm gonna get to pi. Arc cosine of negative one is going to be equal to pi. And so if I were to connect the graph, if I were to connect the dots, it would look something like this. And it's not a coincidence that this looks like a sideways version of one cycle of a cosine function, because when you take the inverse, you're kind of flipping the axes here, and so that's exactly what happened.
The only reason why we can do one cycle is we wanna make sure this is a function and the domain of arc cosine, you can only input between negative one and one, and you can't then repeat it 'cause then if you were to repeat it over and over, you're not going to have a function anymore.
Now, let's do the same thing for arc sine. It's the inverse of sine. So arc sine of negative one is going to be negative pi over two. Arc sine of negative one is negative pi over two. Arc sine of negative one half is going to be negative pi over six. Arc sine of negative one half is going to be negative pi over six, and I didn't quite, negative pi over six would be right here, negative pi over six.
Arc sine of zero is zero, so we go right through the origin, and then arc sine of one half is pi over six. If we input one half, we're gonna get to pi over six. And then last but not least, arc sine of one is pi over two, arc sine of one is pi over two. So what is this gonna look like?
Well, it's gonna look like this. It's gonna look like part of a sideways sine function. Once again, not a coincidence. And for the same reasons, our domain is restricted between negative one and one because if you think about a sine or cosine function, its range is restricted to between one and negative one, and so now we have flipped things 'cause we're looking at the inverse function.
So anyway, this is just to give you an idea of what these look like, why they look like, how you can actually draw that, and that will be helpful as you start to study arc sine and arc cosine a little bit more.
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