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So we know that the sum of the angles of a triangle add up to 180. Now in a right angle, one of the angles is 90 degrees. So that means that the other two must add up to 90. These two add up to 90, plus another 90 is going to be 180 degrees. Or another way to think about it is that the other two non-right angles are going to be complementary. So what plus 32 is equal to 90? Well, 90 minus 32 is 58. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
These two add up to 90, plus another 90 is going to be 180 degrees. Or another way to think about it is that the other two non-right angles are going to be complementary. So what plus 32 is equal to 90? Well, 90 minus 32 is 58. So this right over here is going to be 58 degrees. Well, why is that interesting? Well, we already know what the cosine of 58 degrees is equal to. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, 90 minus 32 is 58. So this right over here is going to be 58 degrees. Well, why is that interesting? Well, we already know what the cosine of 58 degrees is equal to. But let's think about it in terms of ratios of the lengths of sides of this right triangle. Let's just write down SOH CAH TOA. So sine is opposite over hypotenuse. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, we already know what the cosine of 58 degrees is equal to. But let's think about it in terms of ratios of the lengths of sides of this right triangle. Let's just write down SOH CAH TOA. So sine is opposite over hypotenuse. CAH, cosine is adjacent over hypotenuse. TOA, tangent is opposite over adjacent. So we could write down the cosine of 58 degrees, which we already know. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
So sine is opposite over hypotenuse. CAH, cosine is adjacent over hypotenuse. TOA, tangent is opposite over adjacent. So we could write down the cosine of 58 degrees, which we already know. If we think about it in terms of these fundamental ratios, cosine is adjacent over hypotenuse. This is the 58 degree angle. The side that is adjacent to it is, let me do it in this color, is side BC, right over here. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
So we could write down the cosine of 58 degrees, which we already know. If we think about it in terms of these fundamental ratios, cosine is adjacent over hypotenuse. This is the 58 degree angle. The side that is adjacent to it is, let me do it in this color, is side BC, right over here. It's one of the sides of the angle, the side of the angle that is not the hypotenuse. The other side, this over here, is the hypotenuse. So this is going to be the adjacent, the length of the adjacent side, BC, over the length of the hypotenuse. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
The side that is adjacent to it is, let me do it in this color, is side BC, right over here. It's one of the sides of the angle, the side of the angle that is not the hypotenuse. The other side, this over here, is the hypotenuse. So this is going to be the adjacent, the length of the adjacent side, BC, over the length of the hypotenuse. Over the length of the hypotenuse. The length of the hypotenuse, well that is AB. Now let's think about what the sine of 32 degrees would be. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
So this is going to be the adjacent, the length of the adjacent side, BC, over the length of the hypotenuse. Over the length of the hypotenuse. The length of the hypotenuse, well that is AB. Now let's think about what the sine of 32 degrees would be. So the sine of 32 degrees, well sine is opposite over hypotenuse. So now we're looking at this 32 degree angle. What side is opposite? | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
Now let's think about what the sine of 32 degrees would be. So the sine of 32 degrees, well sine is opposite over hypotenuse. So now we're looking at this 32 degree angle. What side is opposite? Well it opens up onto BC. It opens up onto BC, just like that. And what's the hypotenuse? | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
What side is opposite? Well it opens up onto BC. It opens up onto BC, just like that. And what's the hypotenuse? Well we've already, or the length of the hypotenuse, it's AB. It's AB. Notice, the sine of 32 degrees is BC over AB. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
And what's the hypotenuse? Well we've already, or the length of the hypotenuse, it's AB. It's AB. Notice, the sine of 32 degrees is BC over AB. The cosine of 58 degrees is BC over AB. Or another way of thinking about it, the sine of this angle is the same thing as the cosine of this angle. So we could literally write the sine, I'm gonna do that in that pink color, the sine, the sine of 32 degrees is equal to the cosine, cosine of 58 degrees, which is roughly equal to 0.53. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
Notice, the sine of 32 degrees is BC over AB. The cosine of 58 degrees is BC over AB. Or another way of thinking about it, the sine of this angle is the same thing as the cosine of this angle. So we could literally write the sine, I'm gonna do that in that pink color, the sine, the sine of 32 degrees is equal to the cosine, cosine of 58 degrees, which is roughly equal to 0.53. And this is a really, really useful property. The sine of an angle is equal to the cosine of its complement. So we could write this in general terms. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
So we could literally write the sine, I'm gonna do that in that pink color, the sine, the sine of 32 degrees is equal to the cosine, cosine of 58 degrees, which is roughly equal to 0.53. And this is a really, really useful property. The sine of an angle is equal to the cosine of its complement. So we could write this in general terms. We could write that the sine of some angle is equal to the cosine of its complement, is equal to the cosine of 90 minus theta. Think about it. I could have done, I could change this entire problem. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
So we could write this in general terms. We could write that the sine of some angle is equal to the cosine of its complement, is equal to the cosine of 90 minus theta. Think about it. I could have done, I could change this entire problem. Instead of making this the sine of 32 degrees, I could make this the sine of 25 degrees. And if someone gave you the cosine of what's 90 minus 25, if someone gave you the cosine of 65 degrees, then you could think about this as 25. The complement is going to be right over here. | Sine and cosine of complements example Basic trigonometry Trigonometry Khan Academy.mp3 |
Sort the expressions according to their values. You can put any number of cards in a category or leave a category empty. And so we have this diagram right over here. And then we have these cards that have these expressions. And we're supposed to sort these into different buckets. So we're trying to say, well, what is the length of segment AC over the length of segment BC equal to? Which of these expressions is it equal to? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
And then we have these cards that have these expressions. And we're supposed to sort these into different buckets. So we're trying to say, well, what is the length of segment AC over the length of segment BC equal to? Which of these expressions is it equal to? And then we should drag it into the appropriate buckets. So to figure these out, I've actually already redrawn this problem on my little, I guess you'd call it, scratch pad or blackboard, whatever you want to call it. This right over here is that same diagram blown up a little bit. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Which of these expressions is it equal to? And then we should drag it into the appropriate buckets. So to figure these out, I've actually already redrawn this problem on my little, I guess you'd call it, scratch pad or blackboard, whatever you want to call it. This right over here is that same diagram blown up a little bit. Here are the expressions that we need to drag into things. And here are the buckets that we need to see which of these expressions are equal to which of these expressions. So let's first look at this. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
This right over here is that same diagram blown up a little bit. Here are the expressions that we need to drag into things. And here are the buckets that we need to see which of these expressions are equal to which of these expressions. So let's first look at this. The length of segment AC over the length of segment BC. So let's think about what AC is. The length of segment AC. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So let's first look at this. The length of segment AC over the length of segment BC. So let's think about what AC is. The length of segment AC. AC is this right over here. So it's this length right over here in purple over the length of segment BC. Over this length right over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
The length of segment AC. AC is this right over here. So it's this length right over here in purple over the length of segment BC. Over this length right over here. So it's the ratio of the lengths of two sides of a right triangle. This is clearly a right triangle. Triangle ABC. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Over this length right over here. So it's the ratio of the lengths of two sides of a right triangle. This is clearly a right triangle. Triangle ABC. And I could color that in just so you know what triangle I'm talking about. Triangle ABC is this entire triangle that we could focus on. So you could imagine that it's reasonable that the ratio of two sides of a right triangle are going to be the sine of one of its angles. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Triangle ABC. And I could color that in just so you know what triangle I'm talking about. Triangle ABC is this entire triangle that we could focus on. So you could imagine that it's reasonable that the ratio of two sides of a right triangle are going to be the sine of one of its angles. And they give us one of the angles right over here. They give us this angle right over here. You say, well, no, all they did is mark that angle. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So you could imagine that it's reasonable that the ratio of two sides of a right triangle are going to be the sine of one of its angles. And they give us one of the angles right over here. They give us this angle right over here. You say, well, no, all they did is mark that angle. But notice, one arc is here. One arc is here. So anywhere we see only one arc, that's going to be 30 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
You say, well, no, all they did is mark that angle. But notice, one arc is here. One arc is here. So anywhere we see only one arc, that's going to be 30 degrees. So this is 30 degrees as well. You have two arcs here. That's 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So anywhere we see only one arc, that's going to be 30 degrees. So this is 30 degrees as well. You have two arcs here. That's 41 degrees. Two arcs here. This is going to be congruent to that. This over here is going to be 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
That's 41 degrees. Two arcs here. This is going to be congruent to that. This over here is going to be 41 degrees. This is three arcs. They don't tell us how many degrees that is. But this angle with the three arcs is congruent to this angle with the three arcs right over there. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
This over here is going to be 41 degrees. This is three arcs. They don't tell us how many degrees that is. But this angle with the three arcs is congruent to this angle with the three arcs right over there. So anyway, this yellow triangle, triangle ABC, we know the measure of this angle is 30 degrees. And then they give us these two sides. So how do these sides relate to this 30 degree angle? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
But this angle with the three arcs is congruent to this angle with the three arcs right over there. So anyway, this yellow triangle, triangle ABC, we know the measure of this angle is 30 degrees. And then they give us these two sides. So how do these sides relate to this 30 degree angle? Well, side AC is adjacent to it. It's literally one of the sides of the angle that is not the hypotenuse. So let me write that down. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So how do these sides relate to this 30 degree angle? Well, side AC is adjacent to it. It's literally one of the sides of the angle that is not the hypotenuse. So let me write that down. This is adjacent. And what is BC? Well, BC is the hypotenuse of this right triangle. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So let me write that down. This is adjacent. And what is BC? Well, BC is the hypotenuse of this right triangle. It's the side opposite the 90 degrees. So this is the hypotenuse. So what trig function, when applied to 30 degrees, is equal to the adjacent side over the hypotenuse? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, BC is the hypotenuse of this right triangle. It's the side opposite the 90 degrees. So this is the hypotenuse. So what trig function, when applied to 30 degrees, is equal to the adjacent side over the hypotenuse? Let's write down SOH CAH TOA just to remind ourselves. So SOH CAH TOA, sine of an angle is opposite over a hypotenuse. Cosine of an angle is adjacent over a hypotenuse. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So what trig function, when applied to 30 degrees, is equal to the adjacent side over the hypotenuse? Let's write down SOH CAH TOA just to remind ourselves. So SOH CAH TOA, sine of an angle is opposite over a hypotenuse. Cosine of an angle is adjacent over a hypotenuse. So cosine, let's write this down, cosine of 30 degrees is going to be equal to the length of the adjacent side, so that is AC, over the length of the hypotenuse, which is equal to BC. So this right over here is the same thing as the cosine of 30 degrees. So let's drag it in there. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Cosine of an angle is adjacent over a hypotenuse. So cosine, let's write this down, cosine of 30 degrees is going to be equal to the length of the adjacent side, so that is AC, over the length of the hypotenuse, which is equal to BC. So this right over here is the same thing as the cosine of 30 degrees. So let's drag it in there. This is equal to the cosine of 30 degrees. Now let's look at the next one. Cosine of angle DEC. Where is DEC? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So let's drag it in there. This is equal to the cosine of 30 degrees. Now let's look at the next one. Cosine of angle DEC. Where is DEC? So DEC. So that's this angle right over here. I'll put four arcs here so we don't get it confused. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Cosine of angle DEC. Where is DEC? So DEC. So that's this angle right over here. I'll put four arcs here so we don't get it confused. So this is angle DEC. So what is the cosine of DEC? Well, once again, cosine is adjacent over a hypotenuse. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
I'll put four arcs here so we don't get it confused. So this is angle DEC. So what is the cosine of DEC? Well, once again, cosine is adjacent over a hypotenuse. So cosine of angle DEC, the adjacent side to this, well, that's this right over here. You might say, well, isn't this side adjacent? Well, that side, side DE, that is the actual hypotenuse. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, once again, cosine is adjacent over a hypotenuse. So cosine of angle DEC, the adjacent side to this, well, that's this right over here. You might say, well, isn't this side adjacent? Well, that side, side DE, that is the actual hypotenuse. So that's not going to be the adjacent side. So the adjacent side is E. The adjacent side is, I could call it EC. It's the length of segment EC. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, that side, side DE, that is the actual hypotenuse. So that's not going to be the adjacent side. So the adjacent side is E. The adjacent side is, I could call it EC. It's the length of segment EC. And then the hypotenuse is this right over here. It's the length of the hypotenuse. The hypotenuse is side DE or ED, however you want to call it. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
It's the length of segment EC. And then the hypotenuse is this right over here. It's the length of the hypotenuse. The hypotenuse is side DE or ED, however you want to call it. And so the length of it is, we could just write it as DE. Now what is this also equal to? We don't see this choice over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
The hypotenuse is side DE or ED, however you want to call it. And so the length of it is, we could just write it as DE. Now what is this also equal to? We don't see this choice over here. We don't have the ratio EC over DE as one of these choices here. But what we do have is we do get one of the angles here. They give us this 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
We don't see this choice over here. We don't have the ratio EC over DE as one of these choices here. But what we do have is we do get one of the angles here. They give us this 41 degrees. And the ratio of this green side, the length of this green side over this orange side, what would that be in terms of if we wanted to apply a trig function to this angle? Well, relative to this angle, the green side is the opposite side. And the orange side is still the hypotenuse. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
They give us this 41 degrees. And the ratio of this green side, the length of this green side over this orange side, what would that be in terms of if we wanted to apply a trig function to this angle? Well, relative to this angle, the green side is the opposite side. And the orange side is still the hypotenuse. So relative to 41 degrees, so let's write this down. Relative to 41 degrees, this ratio is the opposite over the hypotenuse. It's the cosine of this angle. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
And the orange side is still the hypotenuse. So relative to 41 degrees, so let's write this down. Relative to 41 degrees, this ratio is the opposite over the hypotenuse. It's the cosine of this angle. But it's the sine of this angle right over here. So sine is opposite over hypotenuse. So this is equal to the sine of this angle right over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
It's the cosine of this angle. But it's the sine of this angle right over here. So sine is opposite over hypotenuse. So this is equal to the sine of this angle right over here. It's equal to the sine of 41 degrees. So that is this one right over here, the sine of 41 degrees. So let's drag that into the appropriate bucket. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So this is equal to the sine of this angle right over here. It's equal to the sine of 41 degrees. So that is this one right over here, the sine of 41 degrees. So let's drag that into the appropriate bucket. So let's sine of 41 degrees is the same thing as the cosine of angle DEC. Only have two left. So now we have to figure out what the sine of angle CDA is. So let's see, where is CDA? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So let's drag that into the appropriate bucket. So let's sine of 41 degrees is the same thing as the cosine of angle DEC. Only have two left. So now we have to figure out what the sine of angle CDA is. So let's see, where is CDA? CDA is this entire angle. It's this entire angle right over here. So I could put a bunch of arcs here if I want, just to show that it's different than all the other ones. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So let's see, where is CDA? CDA is this entire angle. It's this entire angle right over here. So I could put a bunch of arcs here if I want, just to show that it's different than all the other ones. So that's that angle right over there. So now we're really dealing with this larger right triangle. Let me highlight that in some. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So I could put a bunch of arcs here if I want, just to show that it's different than all the other ones. So that's that angle right over there. So now we're really dealing with this larger right triangle. Let me highlight that in some. Let me highlight it in this pink color. So we're now dealing with this larger right triangle right over here. We care about the sine of this whole thing. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Let me highlight that in some. Let me highlight it in this pink color. So we're now dealing with this larger right triangle right over here. We care about the sine of this whole thing. Remember, sine is opposite over hypotenuse. Sine is opposite over hypotenuse. So the opposite side is going to be side CA. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
We care about the sine of this whole thing. Remember, sine is opposite over hypotenuse. Sine is opposite over hypotenuse. So the opposite side is going to be side CA. So this is going to be equal to the length of CA over the hypotenuse, which is AD. So that is going to be over AD. Now once again, we don't see that as a choice here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So the opposite side is going to be side CA. So this is going to be equal to the length of CA over the hypotenuse, which is AD. So that is going to be over AD. Now once again, we don't see that as a choice here. But maybe we can express this ratio. Maybe this ratio is a trig function applied to one of the other angles. And they give us one of the angles. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Now once again, we don't see that as a choice here. But maybe we can express this ratio. Maybe this ratio is a trig function applied to one of the other angles. And they give us one of the angles. They give us this angle right over here. I guess we could call this angle DAC. This is 30 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
And they give us one of the angles. They give us this angle right over here. I guess we could call this angle DAC. This is 30 degrees. So relative to this angle, what two sides are we taking the ratio of? We're taking now the ratio of, relative to this angle, the adjacent side over the hypotenuse. So this is the adjacent side over the hypotenuse. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
This is 30 degrees. So relative to this angle, what two sides are we taking the ratio of? We're taking now the ratio of, relative to this angle, the adjacent side over the hypotenuse. So this is the adjacent side over the hypotenuse. What deals with adjacent over hypotenuse? Well, cosine. So this is equal to the cosine of this angle. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So this is the adjacent side over the hypotenuse. What deals with adjacent over hypotenuse? Well, cosine. So this is equal to the cosine of this angle. So this is equal to cosine of 30 degrees. Sine of CDA is equal to the cosine of this angle right over here. So this one is equal to this right over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So this is equal to the cosine of this angle. So this is equal to cosine of 30 degrees. Sine of CDA is equal to the cosine of this angle right over here. So this one is equal to this right over here. So let me drag that in. So this one is equal to, so you can see that I just dragged it in, equal to that. And now we have one left. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So this one is equal to this right over here. So let me drag that in. So this one is equal to, so you can see that I just dragged it in, equal to that. And now we have one left. We have one left. Home stretch. We should be getting excited. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
And now we have one left. We have one left. Home stretch. We should be getting excited. AE over EB. AE, let me use this color. Length of segment AE. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
We should be getting excited. AE over EB. AE, let me use this color. Length of segment AE. That's this length right over here. Let me make that stand out more. Let me do it in this red. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Length of segment AE. That's this length right over here. Let me make that stand out more. Let me do it in this red. This color right over here. That's length of segment AE over length of segment EB. This is EB right over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Let me do it in this red. This color right over here. That's length of segment AE over length of segment EB. This is EB right over here. This is EB. So now we are focused on this triangle, this right triangle right over here. Well, we know the measure of this angle over here. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
This is EB right over here. This is EB. So now we are focused on this triangle, this right triangle right over here. Well, we know the measure of this angle over here. We have double arcs. We have double arcs right over here. And they say this is 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
Well, we know the measure of this angle over here. We have double arcs. We have double arcs right over here. And they say this is 41 degrees. So we have double marks over here. And this is also going to be 41 degrees. So relative to this angle, what ratio is this? | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
And they say this is 41 degrees. So we have double marks over here. And this is also going to be 41 degrees. So relative to this angle, what ratio is this? This is the opposite over the hypotenuse. Opposite over the hypotenuse. This right over here is going to be sine of that angle, sine of 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So relative to this angle, what ratio is this? This is the opposite over the hypotenuse. Opposite over the hypotenuse. This right over here is going to be sine of that angle, sine of 41 degrees. So it's equal to this first one right over there. So let's drag it. So this is going to be equal to sine of 41 degrees. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
This right over here is going to be sine of that angle, sine of 41 degrees. So it's equal to this first one right over there. So let's drag it. So this is going to be equal to sine of 41 degrees. So none of the ones actually ended up being equal to the tangent of 41 degrees. Now let's see if we actually got this right. I hope I did. | Example relating trig function to side ratios Basic trigonometry Trigonometry Khan Academy.mp3 |
So the first thing we have to ask ourselves is what does amplitude even refer to? Well, the amplitude of a periodic function is just half the difference between the minimum and maximum values it takes on. So if I were to draw a periodic function like this, and if we just go back and forth between two, let me draw it a little bit neater, it goes back and forth between two values like that. So between that value and that value, you take the difference between the two, and half of that is the amplitude. Another way of thinking about the amplitude is how much does it sway from its middle position? Right over here, we have y equals negative 1 half cosine of 3x. So what is going to be the amplitude of this? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So between that value and that value, you take the difference between the two, and half of that is the amplitude. Another way of thinking about the amplitude is how much does it sway from its middle position? Right over here, we have y equals negative 1 half cosine of 3x. So what is going to be the amplitude of this? Well, the easy way to think about it is just what is multiplying the cosine function? And you could do the same thing if it was a sine function. We have negative 1 half multiplying it. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So what is going to be the amplitude of this? Well, the easy way to think about it is just what is multiplying the cosine function? And you could do the same thing if it was a sine function. We have negative 1 half multiplying it. So the amplitude in this situation is going to be the absolute value of negative 1 half, which is equal to 1 half. And you might say, well, why do I not care about the sine? Why do I take the absolute value of it? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
We have negative 1 half multiplying it. So the amplitude in this situation is going to be the absolute value of negative 1 half, which is equal to 1 half. And you might say, well, why do I not care about the sine? Why do I take the absolute value of it? Well, the negative just flips the function around. It's not going to change how much it sways between its minimum and maximum position. The other thing is, well, how is it just simply the absolute value of this thing? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Why do I take the absolute value of it? Well, the negative just flips the function around. It's not going to change how much it sways between its minimum and maximum position. The other thing is, well, how is it just simply the absolute value of this thing? And to realize the why, you just have to remember that a cosine function or a sine function varies between positive 1 and negative 1 if it's just a simple function. So this is just multiplying that positive 1 or negative 1. And so if normally the amplitude, if you didn't have any coefficient here, if the coefficient was positive or negative 1, the amplitude would just be 1. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
The other thing is, well, how is it just simply the absolute value of this thing? And to realize the why, you just have to remember that a cosine function or a sine function varies between positive 1 and negative 1 if it's just a simple function. So this is just multiplying that positive 1 or negative 1. And so if normally the amplitude, if you didn't have any coefficient here, if the coefficient was positive or negative 1, the amplitude would just be 1. Now you're changing it or you're multiplying it by this amount. So the amplitude is 1 half. Now let's think about the period. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And so if normally the amplitude, if you didn't have any coefficient here, if the coefficient was positive or negative 1, the amplitude would just be 1. Now you're changing it or you're multiplying it by this amount. So the amplitude is 1 half. Now let's think about the period. So the first thing I want to ask you is what does the period of a cyclical function, even or a periodic function, I should say, what does the period of a periodic function even refer to? Well, let me draw some axes on this function right over here. Let's say that this right over here is the y-axis. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Now let's think about the period. So the first thing I want to ask you is what does the period of a cyclical function, even or a periodic function, I should say, what does the period of a periodic function even refer to? Well, let me draw some axes on this function right over here. Let's say that this right over here is the y-axis. That's the y-axis. And let's just say, for the sake of argument, this is the x-axis right over here. So the period of a periodic function is the length of the smallest interval that contains exactly one copy of the repeating pattern of that periodic function. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Let's say that this right over here is the y-axis. That's the y-axis. And let's just say, for the sake of argument, this is the x-axis right over here. So the period of a periodic function is the length of the smallest interval that contains exactly one copy of the repeating pattern of that periodic function. So what do they mean here? Well, what's repeating? So we go down and then up, just like that. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So the period of a periodic function is the length of the smallest interval that contains exactly one copy of the repeating pattern of that periodic function. So what do they mean here? Well, what's repeating? So we go down and then up, just like that. Then we go down and then we go up. So in this case, the length of the smallest interval that contains exactly one copy of the repeating pattern, this could be one of the smallest repeating patterns. And so this length between here and here would be one period. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So we go down and then up, just like that. Then we go down and then we go up. So in this case, the length of the smallest interval that contains exactly one copy of the repeating pattern, this could be one of the smallest repeating patterns. And so this length between here and here would be one period. Then we could go between here and here is another period. And there's multiple. This isn't the only pattern that you could pick. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And so this length between here and here would be one period. Then we could go between here and here is another period. And there's multiple. This isn't the only pattern that you could pick. You could say, well, I'm going to define my pattern starting here, going up, and then going down, like that. So you could say, that's my smallest length. And then you would see that, OK, well, if you go in the negative direction, the next repeating version of that pattern is right over there. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
This isn't the only pattern that you could pick. You could say, well, I'm going to define my pattern starting here, going up, and then going down, like that. So you could say, that's my smallest length. And then you would see that, OK, well, if you go in the negative direction, the next repeating version of that pattern is right over there. But either way, you're going to get the same length that it takes to repeat that pattern. So given that, what is the period of this function right over here? Well, to figure out the period, we just take 2 pi and divide it by the absolute value of the coefficient right over here. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And then you would see that, OK, well, if you go in the negative direction, the next repeating version of that pattern is right over there. But either way, you're going to get the same length that it takes to repeat that pattern. So given that, what is the period of this function right over here? Well, to figure out the period, we just take 2 pi and divide it by the absolute value of the coefficient right over here. So we divide it by the absolute value of 3, which is just 3. So we get 2 pi over 3. Now, we need to think about why does this work. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Well, to figure out the period, we just take 2 pi and divide it by the absolute value of the coefficient right over here. So we divide it by the absolute value of 3, which is just 3. So we get 2 pi over 3. Now, we need to think about why does this work. Well, if you think about just a traditional cosine function or a traditional sine function, it has a period of 2 pi. If you think about the unit circle, 2 pi, if you start at 0, 2 pi radians later, you're back to where you started. 2 pi radians, another 2 pi, you're back to where you started. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Now, we need to think about why does this work. Well, if you think about just a traditional cosine function or a traditional sine function, it has a period of 2 pi. If you think about the unit circle, 2 pi, if you start at 0, 2 pi radians later, you're back to where you started. 2 pi radians, another 2 pi, you're back to where you started. If you go in the negative direction, you go negative 2 pi, you're back to where you started. For any angle here, if you go 2 pi, you're back to where you were before. You go negative 2 pi, you're back to where you were before. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
2 pi radians, another 2 pi, you're back to where you started. If you go in the negative direction, you go negative 2 pi, you're back to where you started. For any angle here, if you go 2 pi, you're back to where you were before. You go negative 2 pi, you're back to where you were before. So the periods for these are all 2 pi. And the reason why this makes sense is that this coefficient makes you get to 2 pi or negative, in this case, 2 pi, it's going to make you get to 2 pi all that much faster. And so your period is going to be a lower number. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
You go negative 2 pi, you're back to where you were before. So the periods for these are all 2 pi. And the reason why this makes sense is that this coefficient makes you get to 2 pi or negative, in this case, 2 pi, it's going to make you get to 2 pi all that much faster. And so your period is going to be a lower number. It takes less length. You're going to get to 2 pi 3 times as fast. Now, you might say, well, why are you taking the absolute value here? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And so your period is going to be a lower number. It takes less length. You're going to get to 2 pi 3 times as fast. Now, you might say, well, why are you taking the absolute value here? Well, if this was a negative number, it would get you to negative 2 pi all that much faster. But either way, you're going to be completing one cycle. So with that out of the way, let's visualize these two things. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Now, you might say, well, why are you taking the absolute value here? Well, if this was a negative number, it would get you to negative 2 pi all that much faster. But either way, you're going to be completing one cycle. So with that out of the way, let's visualize these two things. Let's actually draw negative 1 half cosine of 3x. So let me draw my axes here, my best attempt. So this is my y-axis. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So with that out of the way, let's visualize these two things. Let's actually draw negative 1 half cosine of 3x. So let me draw my axes here, my best attempt. So this is my y-axis. This is my x-axis. And then let me draw some. So this is 0 right over here. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So this is my y-axis. This is my x-axis. And then let me draw some. So this is 0 right over here. x is equal to 0. And let me draw x is equal to positive 1 half. I'll draw it right over here. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So this is 0 right over here. x is equal to 0. And let me draw x is equal to positive 1 half. I'll draw it right over here. So x is equal to positive 1 half. And we haven't shifted this function up or down any. Then if we wanted to, we could add a constant out here, outside of the cosine function. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
I'll draw it right over here. So x is equal to positive 1 half. And we haven't shifted this function up or down any. Then if we wanted to, we could add a constant out here, outside of the cosine function. But this is positive 1 half. Or we could just write that as 1 half. And then down here, let's say that this is negative 1 half. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
Then if we wanted to, we could add a constant out here, outside of the cosine function. But this is positive 1 half. Or we could just write that as 1 half. And then down here, let's say that this is negative 1 half. And so let me draw that bound. I'm just drawing these dotted lines, so it'll become easy for me to draw. And what happens when this is 0? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And then down here, let's say that this is negative 1 half. And so let me draw that bound. I'm just drawing these dotted lines, so it'll become easy for me to draw. And what happens when this is 0? Well, cosine of 0 is 1. But we're going to multiply it by negative 1 half. So it's going to be negative 1 half right over here. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And what happens when this is 0? Well, cosine of 0 is 1. But we're going to multiply it by negative 1 half. So it's going to be negative 1 half right over here. And then it's going to start going up. It can only go in that direction. It's bounded. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So it's going to be negative 1 half right over here. And then it's going to start going up. It can only go in that direction. It's bounded. It's going to start going up. Then it'll come back down. And then it will get back to that original point right over here. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
It's bounded. It's going to start going up. Then it'll come back down. And then it will get back to that original point right over here. And the question is, what is this distance? What is this length? What is this length going to be? | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And then it will get back to that original point right over here. And the question is, what is this distance? What is this length? What is this length going to be? Well, we know what its period is. It's 2 pi over 3. It's going to get to this point 3 times as fast as a traditional cosine function. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
What is this length going to be? Well, we know what its period is. It's 2 pi over 3. It's going to get to this point 3 times as fast as a traditional cosine function. So this is going to be 2 pi over 3. And then if you give it another 2 pi over 3, it's going to get back to that same point again. So if you go another 2 pi over 3, so in this case, you've now gone 4 pi over 3. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
It's going to get to this point 3 times as fast as a traditional cosine function. So this is going to be 2 pi over 3. And then if you give it another 2 pi over 3, it's going to get back to that same point again. So if you go another 2 pi over 3, so in this case, you've now gone 4 pi over 3. 4 pi over 3, you've completed another cycle. So that length right over there is the period. And you could also do the same thing in the negative direction. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
So if you go another 2 pi over 3, so in this case, you've now gone 4 pi over 3. 4 pi over 3, you've completed another cycle. So that length right over there is the period. And you could also do the same thing in the negative direction. So this right over here would be negative 2 pi over 3. And to visualize the amplitude, you see that it can go 1 half. Well, there's two ways to think about it. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |
And you could also do the same thing in the negative direction. So this right over here would be negative 2 pi over 3. And to visualize the amplitude, you see that it can go 1 half. Well, there's two ways to think about it. The difference between the maximum and the minimum point is 1. Half of that is 1 half. Or you could say that it's going 1 half in magnitude. | Example Amplitude and period Graphs of trig functions Trigonometry Khan Academy.mp3 |