Learn about The unit circle definition of trigonometric function Video

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Learn about The unit circle definition of trigonometric function

We are now going to study the unit circle a little bit more and see how it extends I guess we could say the traditional soh-cah-toa definitions and functions and how we can actually use it to solve for angles that the soh-cah-toa definition of the trig functions actually does not help us with. So, let’s just, as a review remember what soh-cah-toa told us. It told us or with that definition alright up here in this corner I do not know how. So I do not want to get confusing so I do not want to write over too much, soh-cah-toa and that told us that sine if we have a right angle that the sine of an angle in the right angle is equal to the opposite side over the hypotenuse, the cosine of an angle is equal to the adjacent side over hypotenuse and the tangent side is equal to the opposite over the adjacent side. And this worked fine for us but if you think about what happens when that angle approaches 90 degrees because you cannot have two 90-degree angles in a right angle or what happens when that angle is greater than 90 degrees or what if it went negative and that is why if you remember from the previous video why we needed the unit circle definition. So, let us review the unit circle definition. I actually got this unit circled. I think I got it from wikipedia but I want to give due credit to whoever I did get it for this drawing of the unit circle. But the unit circle kind of extends that soh-cah-toa definition. It tells us if we have a unit circle and this is a picture of a unit circle here, a unit circle is just a circle centered at the origin, centered at the point (0,0) and it has a radius of one so it intersects the x-axis at (1,0) and (-1,0), intersects the y-axis at (0,1) and (0,-1). If we have a unit circle, let us just say let us start with the cosine of theta, we define the cosine of theta as let us take an angle that is between two radiuses in this unit circle. And one radius is going to be the positive x-axis between 0 and 1 so one radius is going to be this line here and then we have the angle is the angle between that. You can kind of view that as the base radius and some other radius so let us say this case right here and this would be our angle. The unit circle definition tells us that the cosine of this angle is equal to the x-coordinate where this radius intersects the unit circle and that the sine of this function is equal to the y-coordinate where at this point intersects the unit circle. So for example, in this case if you can read behind my line, this is 30 degrees equal pi over 6 so this angle right here is 30 degrees or π/6 radians, π/6 radians. And what this definition tells us is that the sine of 30 degrees is ½ and that the cosine of 30 degrees is square root of 3/2 and what I want to show you is that this unit circle definition actually coincides with our soh-cah-toa definition but then it extends it so let us see how we can get from that soh-cah-toa definition to this unit circle definition and why they are actually consistent with each other. Okay, I think I am all set. So, let us go back to that theta. Let us say that this is the theta. And as we said this angle is 30 degrees or π/6. Let us drop a line from that point to the x-axis. And as we see this line is perpendicular so this is a 90-degree angle. And if this is a 30-degree angle here, this is 30, alright theta equals 30, that says theta equals 30. This is 30, this is 90, what is this angle? Well, this is a 60-degree angle right? Because they add up to 180, so this is a 30-60-90 triangle, interesting. And what do you remember about 30-60-90 triangles? Well, the side opposite, the 30-degree side is ½ the length of the hypotenuse. I hope you remember that. I do not want to confuse you too much. So, this is the side opposite the 30-degree side, right? And what is the hypotenuse? This is the hypotenuse and what is the length of this hypotenuse? Well, it is one because this is a unit circle and this is a radius of the unit circle so the length of this hypotenuse is one and the length of this side which is opposite the 30-degree angle is going to be ½ right? And I am just using the 30-60-90 triangles that we have done previous videos on. And what is the side opposite of the 60-degree side? Well, once again it is square root of 3/2 times the hypotenuse, so it’s square root of 3 over 2, right? So, we can figure out that this side is square root of 3/2 and that this side is ½. So, a couple of things we can figure out. Just by looking at this, we can immediately say, “Well, what is the coordinates of this point?” Well, its x-coordinate is right here, right? Its x-coordinate would be square root of 3/2, that is this right here, this distance and its y-coordinate would be the length of this side of the right triangle or ½ and there we have it right here. The x-coordinate is square root of 3/2 and the y-coordinate is ½. And now, what I want to show you is why this x-coordinate can be taken as a cosine of theta and why this y-coordinate can be taken as a sine of theta but what does soh-cah-toa tell us? Well, let us start with the cosine, so soh-cah-toa. Cosine is adjacent over hypotenuse right? Cosine is equal to adjacent over hypotenuse. Well, in this triangle I just drew, what is the adjacent side? To this angle right because we are trying to figure out the cosine of this angle, it’s 30 degrees. Well, the adjacent side to this angle is of course this side right? So, adjacent is square root of 3/2. We’ve figured that out just now. And what is the hypotenuse? Well, the hypotenuse is this side, right, which has a length one because it is a unit circle and that is the radius of it. So, the cosine of this angle using the soh-cah-toa definition is square root of 3, the adjacent side over the hypotenuse 1, so square root of 3/2 over 1 which is just square root of 3/2 which was the same thing as the x-coordinate. Similarly, we could look at sine equals opposite over hypotenuse but what is the opposite side? It is ½ and the hypotenuse is 1 here so the sine is just ½ over 1 and so we have it here. And so that is why the unit circle definition is not kind of a replacing definition for the soh-cah-toa definition. It is really just an extension that it allows us, I mean for a 30-degree soh-cah-toa worked, for 45 degrees soh-cah-toa worked, for 60 degrees it would work but once you get to 90 it becomes a little bit more difficult right if you use traditional soh-cah-toa and you try to draw a right triangle that has two 90-degree angles in it because you could not. And especially once you get to angles that are larger than 90 or angles that actually could even go negative. It is not drawn here in the unit circle but 330 degrees is the same thing as -30 degrees because you could go either way in the circle and you could keep going around the circle, you could figure out the sine or the cosine of a million degrees if you just keep going around the circle. So hopefully, this gives you a sense of the unit circle definition of the sine and cosine functions and of course the tangent function is always just the sine over the cosine or the y over the x and you can use the unit circle definition for that as well. And I will leave it for you as an exercise to try to drive all of these other values using this unit circle and using what you already know about 30-60-90 triangles and what you already know about 45-45-90 triangles or what you already know about the Pythagorean Theorem and you should be able to figure out all of these values going around the unit circle. Hopefully, that was helpful. See you soon!

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We are now going to study the unit circle a little bit more and see how it extends I guess we could say the traditional soh-cah-toa definitions and functions and how we can actually use it to solve for angles that the soh-cah-toa definition of the trig functions actually does not help us with. So, let’s just, as a review remember what soh-cah-toa told us. It told us or with that definition alright up here in this corner I do not know how. So I do not want to get confusing so I do not want to write over too much, soh-cah-toa and that told us that sine if we have a right angle that the sine of an angle in the right angle is equal to the opposite side over the hypotenuse, the cosine of an angle is equal to the adjacent side over hypotenuse and the tangent side is equal to the opposite over the adjacent side. And this worked fine for us but if you think about what happens when that angle approaches 90 degrees because you cannot have two 90-degree angles in a right angle or what happens when that angle is greater than 90 degrees or what if it went negative and that is why if you remember from the previous video why we needed the unit circle definition.

So, let us review the unit circle definition. I actually got this unit circled. I think I got it from wikipedia but I want to give due credit to whoever I did get it for this drawing of the unit circle. But the unit circle kind of extends that soh-cah-toa definition. It tells us if we have a unit circle and this is a picture of a unit circle here, a unit circle is just a circle centered at the origin, centered at the point (0,0) and it has a radius of one so it intersects the x-axis at (1,0) and (-1,0), intersects the y-axis at (0,1) and (0,-1). If we have a unit circle, let us just say let us start with the cosine of theta, we define the cosine of theta as let us take an angle that is between two radiuses in this unit circle. And one radius is going to be the positive x-axis between 0 and 1 so one radius is going to be this line here and then we have the angle is the angle between that. You can kind of view that as the base radius and some other radius so let us say this case right here and this would be our angle.

The unit circle definition tells us that the cosine of this angle is equal to the x-coordinate where this radius intersects the unit circle and that the sine of this function is equal to the y-coordinate where at this point intersects the unit circle. So for example, in this case if you can read behind my line, this is 30 degrees equal pi over 6 so this angle right here is 30 degrees or π/6 radians, π/6 radians. And what this definition tells us is that the sine of 30 degrees is ½ and that the cosine of 30 degrees is square root of 3/2 and what I want to show you is that this unit circle definition actually coincides with our soh-cah-toa definition but then it extends it so let us see how we can get from that soh-cah-toa definition to this unit circle definition and why they are actually consistent with each other.

Okay, I think I am all set. So, let us go back to that theta. Let us say that this is the theta. And as we said this angle is 30 degrees or π/6. Let us drop a line from that point to the x-axis. And as we see this line is perpendicular so this is a 90-degree angle. And if this is a 30-degree angle here, this is 30, alright theta equals 30, that says theta equals 30. This is 30, this is 90, what is this angle? Well, this is a 60-degree angle right? Because they add up to 180, so this is a 30-60-90 triangle, interesting.

And what do you remember about 30-60-90 triangles? Well, the side opposite, the 30-degree side is ½ the length of the hypotenuse. I hope you remember that. I do not want to confuse you too much. So, this is the side opposite the 30-degree side, right? And what is the hypotenuse? This is the hypotenuse and what is the length of this hypotenuse? Well, it is one because this is a unit circle and this is a radius of the unit circle so the length of this hypotenuse is one and the length of this side which is opposite the 30-degree angle is going to be ½ right? And I am just using the 30-60-90 triangles that we have done previous videos on. And what is the side opposite of the 60-degree side? Well, once again it is square root of 3/2 times the hypotenuse, so it’s square root of 3 over 2, right? So, we can figure out that this side is square root of 3/2 and that this side is ½.

So, a couple of things we can figure out. Just by looking at this, we can immediately say, “Well, what is the coordinates of this point?” Well, its x-coordinate is right here, right? Its x-coordinate would be square root of 3/2, that is this right here, this distance and its y-coordinate would be the length of this side of the right triangle or ½ and there we have it right here. The x-coordinate is square root of 3/2 and the y-coordinate is ½. And now, what I want to show you is why this x-coordinate can be taken as a cosine of theta and why this y-coordinate can be taken as a sine of theta but what does soh-cah-toa tell us?

Well, let us start with the cosine, so soh-cah-toa. Cosine is adjacent over hypotenuse right? Cosine is equal to adjacent over hypotenuse. Well, in this triangle I just drew, what is the adjacent side? To this angle right because we are trying to figure out the cosine of this angle, it’s 30 degrees. Well, the adjacent side to this angle is of course this side right? So, adjacent is square root of 3/2. We’ve figured that out just now. And what is the hypotenuse? Well, the hypotenuse is this side, right, which has a length one because it is a unit circle and that is the radius of it. So, the cosine of this angle using the soh-cah-toa definition is square root of 3, the adjacent side over the hypotenuse 1, so square root of 3/2 over 1 which is just square root of 3/2 which was the same thing as the x-coordinate.

Similarly, we could look at sine equals opposite over hypotenuse but what is the opposite side? It is ½ and the hypotenuse is 1 here so the sine is just ½ over 1 and so we have it here. And so that is why the unit circle definition is not kind of a replacing definition for the soh-cah-toa definition. It is really just an extension that it allows us, I mean for a 30-degree soh-cah-toa worked, for 45 degrees soh-cah-toa worked, for 60 degrees it would work but once you get to 90 it becomes a little bit more difficult right if you use traditional soh-cah-toa and you try to draw a right triangle that has two 90-degree angles in it because you could not. And especially once you get to angles that are larger than 90 or angles that actually could even go negative. It is not drawn here in the unit circle but 330 degrees is the same thing as -30 degrees because you could go either way in the circle and you could keep going around the circle, you could figure out the sine or the cosine of a million degrees if you just keep going around the circle.

So hopefully, this gives you a sense of the unit circle definition of the sine and cosine functions and of course the tangent function is always just the sine over the cosine or the y over the x and you can use the unit circle definition for that as well. And I will leave it for you as an exercise to try to drive all of these other values using this unit circle and using what you already know about 30-60-90 triangles and what you already know about 45-45-90 triangles or what you already know about the Pythagorean Theorem and you should be able to figure out all of these values going around the unit circle. Hopefully, that was helpful. See you soon!

Transcription by: Scribe4you Transcription Services