WEBVTT mathematics/trigonometry/murray
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Hi these are the trigonometry lectures for educator.com and today we're talking about computations of inverse trigonometric functions.
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In the previous lecture, we learned the definitions and we practiced a little bit with arcsin, arccos, and arctan, you might want to review those a little bit before you go through this lecture.
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I'm assuming now that you know a little bit about the definitions of arcsin, arccos, and arctan, and we'll practice using them and working them out for some common values today.
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The key thing to remember here is where these functions are defined and what kinds of values you're going to get from them.
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Arcsin, remember you start out with the number between -1 and 1, and you always get an answer between -π/2 and π/2.
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It's very helpful if you remember the unit circle there.
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Arcsin always gives you an angle in the fourth and the first quadrant between -π/2, 0, and π/2.
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You're looking for angles in that range that have a particular sine.
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Arccos, also between -1 and 1, produces an answer between 0, and π.
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Again, it's helpful to draw the unit circle and keep that in mind.
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There's 0, π/2, and π.
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You're trying to find angles between 0 and π that have a given cosine.
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Arctan, you can find the arctan of any number.
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Again, you're trying to find an angle between -π/2 and 0, and π/2 that has a given tangent.
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I say exclusive here because you would never actually give an answer of -π/2 or π/2 for arctan because arctan never actually hits those values.
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If you think of that coming from the other direction, we can't talk about the tangent of π/2 or -π/2, because those involve divisions by 0.
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When we're talking about arctan, we'll never get -π/2 or π/2 as an answer.
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Let's practice finding some common values.
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Here's some common values that we should be able to figure out arc sines of.
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Let me start by drawing my unit circle.
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There's -π/2 and 0, and π/2, remember, our answers always going to be in that range.
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Let me just graph those common values and see what angles they correspond to.
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I'll make a little chart here.
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x and arcsin(x), so we got -1, negative root 3 over 2, negative root 2 over 2, -1/2, 0, 1/2, root 2 over 2, root 3 over 2, and 1.
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Remember, sin(x), sine is the y-coordinate.
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I'm looking for an angle that has y-coordinate of -1 to start with, so I want to find an angle that has y-coordinate down there at -1, and that's clearly -π/2.
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That's the answer.
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Negative root 3 over 2, that's an angle down there, so the angle that has sine of negative root 3 over 2, must be -π/3.
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Negative root 2 over 2, that's the one right there, so that's a -π/4.
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-1/2, the y-coordinate -1/2, is right there, that's -π/3.
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Arcsin(0), what angle has sin(0)?
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Well, it's 0.
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What angle has sin(1/2)?
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Well, what angle has vertical y-coordinate 1/2?
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That's π/3.
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Root 2 over 2, that's our 45-degree angle, also known as π/4.
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Root 3 over 3, that's our 60-degree angle, also known as π/3.
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Finally, we know that the sin(π/2) is 1, so the arcsin(1)=π/2.
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In each case, it's a matter of looking at the value and thinking, "Okay, that's my y-coordinate, where am I on the unit circle?"
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"What angle between -π/2 and π/2 has sine equal to that value?"
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Of course, if you know your values of sine very well, then it's not too hard to figure out the arcsin function.
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You really don't need to memorize this.
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You just need to know your common values for sin(x) very well, and to know when they're positive or negative.
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Then you can figure out the values for arcsin(x).
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In our second problem here, we're asked to find which of the arcsin, acrccos, and arctan functions are odd, even or neither.
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It's really, the key to thinking about this one is probably to think about the graphs and not so much to think about the original definitions of odd or even.
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Let me write down the important properties to remember here.
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An odd function has rotational symmetry around the origin.
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The way I remember that is that 3 is an odd number and x³, y=x³ has rotational symmetry around the origin.
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Even functions have mirror symmetry across the y-axis and the way I remember that is that 2 is an even number, and the graph of y=x² has mirror symmetry across the y-axis.
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That's how I remember the pictures for odd or even functions.
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Now, let me draw the graphs of arcsin and arccos, and arctan, and we'll just test them out.
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Arcsin, remember you take a piece of the sine graph, there's sin(x) or sin(θ).
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Arcsin, I'll draw this one in blue.
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It's the reflection of that graph in the y=x line, that's arcsin(x) in blue.
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Now, if you check that out, that has rotational symmetry around the origin.
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So, arcsin(x) is odd.
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Let's take a look at cosine and arccos.
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Cosine, remember, you got to snip off a piece of cosine graph that will make arccos into a function.
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There's cos(θ), and now in blue, I'll graph arccos.
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There's arccos(x) in blue.
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Now, that graph is neither mirror symmetric across the y-axis nor is it rotationally symmetric around the origin.
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So, it's not odd or even, which is a little bit surprising, because even though if you remember cos(θ) is even.
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It turns out that arccos(x) is not odd or even, even though cos(θ) was an even function.
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Finally, arctan, let me start out by drawing the tangent graph, or at least the piece of it that we're going to snip off.
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It kind of looks like the graph of y=x³, but it's not the same as the graph of y=x³.
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One big difference is that tan(x) has asymptotes at π/2 and -π/2, and of course y=x³ has no asymptotes at all.
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What I just graphed here is tan(θ) and then in blue I'll graph arctan(θ).
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I'm flipping it across the line y=x.
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It also has asymptotes neither horizontal asymptotes.
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That blue graph is arctan(x), and if you look at that, that is rotationally symmetric around the origin.
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So that's also an odd function.
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This problem is really kind of testing whether you know the graphs of arcsin, arccos, and arctan look like.
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If you don't remember those, then you go back to sine, cosine and tangent, and you snip off the important pieces of those graphs, and you flip them around y=x to get the graphs of arcsin, arccos, and arctan.
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Those are the graphs that I have in blue here, arcsin, arccos, and arctan.
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The other thing that this problem is really testing is whether you remember the graphical characterizations of odd and even functions.
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If you know that odd functions have rotational symmetry around the origin, even functions have mirror symmetry across the y-axis, it's easy to check these graphs to just look at them and see whether they have the right kind of symmetry.
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Of course, what you find out is that arcsin(x) has rotational symmetry, arccos(x) doesn't have either one, arctan(x) also has rotational symmetry.
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For our third example here, we're trying to find arccos of the following list of common values.
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Again, it's useful to start with a unit circle here.
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Once you start with a unit circle, remember that with arccos, you're looking for values between 0 and π.
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Arccos is always between 0 and π.
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We're looking for angles between 0 and π that have cosines equal to this list of values.
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I'll make a little chart here.
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-1, negative root 3 over 2, negative root 2 over 2, -1/2, 0, 1/2, root 2 over 2, root 3 over 2 and 1.
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Remember, cosine is the x-value.
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I'm going to draw each one of these values as an x-value, as an x-coordinate, and then I'll see what angle has that particular cosine.
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So, -1, the x-coordinate of -1 is over here, clearly that's π, that's the angle π.
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Negative root 3 over 2, I'll draw that as the x-coordinate, and that angle is 5π/6.
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Negative root 2 over 2, I'll draw that as the x-coordinate, I know that's a 45-degree angle, so that's 3π/4, that's the arccos of negative root 2 over 2
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-1/2, if we draw that as the x-coordinate, that's a 30-60-90 triangle, that's 2π/3.
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What angle has cos(0)?
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That means what angle has x-coordinate 0, that's π/2.
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What angle has cos(1/2)?
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Again, a 30-60-90 triangle, that must be π/3.
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Root 2 over 2, that's a 45-degree angle, so that's π/4.
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Root 3 over 2, that's a 30-60-90 angle again, that's π/6.
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Finally, what angle has x-coordinate 1?
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That's just 0.
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The trick here is remembering your common values of cosine on the unit circle.
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I know all the common values of cosine on the unit circle very well because I remember my 30-60-90 triangles, and I remember my 45-45-90 triangles.
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I know which ones are positive and which ones are negative.
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Finally, I remember that arccos is always between 0 and π.
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So, I'm looking for angles between 0 and π that have these cosines.
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These are the angles that have the right cosines and are in the right range.
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We'll try some more examples of these later.