Now let's talk about the unit circle. So what exactly is the unit circle? The unit circle is a circle with a radius of 1. So if we draw a ray at a 45 degree angle, and let's say if we turn it into a triangle, the hypotenuse of that triangle will be equal to 1. At a 45 degree angle, the x and y portion of the triangle have the same value. So therefore, the point occurs at this value, root 2, comma 2, and the x and y values are the same.
Now you want to know these values because let's say if you wish to find sine of 45 degrees. Sine is equal to, let's say this angle is theta. Sine theta is equal to the y value of this point in the unit circle. Cosine theta is equal to the x value. of any terminal point on a unicircle.
So keep in mind, for the unicircle, the radius is always 1. And when r is 1, cosine is equal to x, and sine is equal to y. So, if we want to find the value of sine 45, it's simply equal to the y coordinate of the point that is associated with 45 degrees. So this is going to be the square root of 2 divided by 2. And you need to know this because it helps you to evaluate sine and cosine functions based on the unicircle.
Now let's talk about some other common values on a unicircle. I'm going to focus mostly on quadrant 1. Because if you know the values in quadrant 1, you could use that to find everything else. There's three common values you need to know.
30, 45, and 60. At a 30 degree angle, the x value is root 3 over 2. And the y value is 1 over 2. At a 45 degree angle... We've covered this one already. It's the square root of 2, comma, over 2, comma, square root 2, over 2. The x and y values are the same. Now at 60... The x value is 1 half, the y value is root 3 over 2. As you can see, some of these values repeat.
At 0 degrees, and at 90, at 0 it's going to be 1 comma 0, and at 90, it's 0 comma 1. At 180, x is negative, y is 0. And at 270, y is negative, but x is 0. So x is 0, y is negative 1. So let's say if you want to evaluate cosine of 270 degrees, you would choose the x value. It would equal 0. Now let's try some other examples. Based on the unit circle, go ahead and evaluate these trig functions. Find the value of sine of 60 degrees, and also find the value of cosine 180. And in addition, find the value of...
sine 30 so sine 60 at 60 degrees look for the y value it's the square root of 3 divided by 2 So that's the value of sine 60 cosine 180 Choose the x value. That's a with the 180 angle and so cosine 180 is negative 1 and sine 30 choose the y value sine 30 is 1 half so if you have access to the unit circle or if you committed to memory, you can easily evaluate any sine or cosine function. Now let's say if we want to find the value of sine 135, cosine 225, and sine of 315, because we didn't have those angles in the circle that we drew.
So how can we find these values without actually having to memorize the entire unit circle? So remember, you only need to know the first quadrant, and also the values at the x and y axis. If you have that, you can figure out everything else.
So let's start with a 45 degree angle. At 45. We said that this correlates to a point that's equal to the square root of 2 over 2, and the x and y values are the same. Now, 45 and 135 are similar. An angle of 135 creates the same reference angle as 45. So this is 135, measured from the positive x-axis.
But it creates a reference angle of 45 with the negative x-axis. Therefore... The x and y values are going to be very similar.
At 135, everything is the same except x. x is negative in quadrant 2, but y is positive. But the numbers are still the same.
Square root 2 over 2, that hasn't changed. The only thing that changes is the sign. So if we wish to evaluate sine of 135, we need to use the y value.
So it's equal to positive square root 2 divided by 2. Now, at 225, it also forms a reference angle of 45. So therefore, these values will be the same. Only the signs will change. So at an angle of 225, These are the coordinates. It's negative square root 2 divided by 2, comma negative square root 2 divided by 2. In quadrant 3, both x and y are negative. So therefore, if we wish to evaluate cosine of 200...
25 degrees it's going to be the x value so it's going to be negative square root 2 divided by 2 now let's go over 315 which also have a reference angle of 45 so at 315 x is positive, but y is negative in quadrant 4. And we're looking for the y value for sine 315. So therefore, it's negative root 2 over 2. So that's why, if you know the angle in quadrant 1, you don't have to memorize the values for quadrant 2, 3, and 4. You can figure it out based on the symmetry of the graph. And I'll give you some more examples of this. Let's find the value of sine pi over 4, I mean pi over 3, cosine 2 pi over 3. sine 4 pi over 3 and also cosine 5 pi over 3 Now if you recall pi is equal to 180 degrees so pi divided by 3 180 over 360 So pi over 3 corresponds to an angle of 60 degrees Now, this is the one that we need to know. At pi over 3, or 60 degrees, you need to know that the point that it corresponds to is this one.
x is 1 half, but y is the square root of 3 over 2. Now, from this value, you can find the other four values. So I'm just going to write this in radians, pi over 3. 2 pi over 3. is in quadrant 2. And then we have 4 pi over 3 and 5 pi over 3. All of these share the same reference angle of 60, or pi over 3. The reference angle is between the terminal side and the x-axis. So here, they all have the same reference angle. Now let's write the values.
At 2 pi over 3, x is negative, but y is positive. And so understanding this process can help you to memorize the entire unit circle. But you really need to know the values in quadrant 1. After that, for quadrants 2, 3, and 4, just change the sign accordingly. The x values are negative in quadrants 2 and 3. That's these two quadrants. And y is negative in 3 and 4. Now, in quadrant 3, both x and y are negative.
So, this is going to be negative 1 half, negative 3 over 2. Everything is based on this value. In quadrant 4, x is positive, but y is negative. So, now we can... evaluate sine pi over 3. So pi over 3, we need to use the y value for sine, so this is equal to the square root of 3 divided by 2. Now cosine 2 pi over 3, we need to use the x value at 2 pi over 3, so that's negative 1 half.
Now sine 4 pi over 3, and we need to use the y value, so that's negative root 3 divided by 2. And finally, for cosine 5 pi over 3, we need to use the x value, which means it's positive 1 half. So now you know how to evaluate trigonometric functions using the unit circle. Go ahead and find these values.
Sine pi over 6. Cosine 5 pi over 6. cosine 7 pi over 6, and sine 11 pi over 6. So pi over 6, that's 180 divided by 6, so that is equal to 30 degrees. So feel free to pause the video and use what we've talked about to evaluate those functions. So let's draw a circle. So this is going to be pi over 6, 5 pi over 6, which is 5 times 30, that's 150. 7 pi over 6 is 210. 11 pi over 6 is 330. Now at pi over 6, x is equal to root 3 divided by 2, but y is 1 half.
At 5 pi over 6, x will be negative, y is positive. At 7 pi over 6, both x and y are negative. And at 11 pi over 6 in quadrant 4, x is positive, but y is negative.
Now, let's go ahead and evaluate sine pi over 6. So, at pi over 6, we need to use the y value to evaluate sine. So, sine pi over 6, or sine 30, is positive 1 half. Now, cosine 5 pi over 6 is positive 1 half. six we need to use the x value so cosine 5 pi over 6 is equal to negative square root 3 divided by 2 cosine is always negative in quadrant 2 and quadrant 3 cosine is negative as well.
But cosine 7 pi over 6 is going to be the same. Negative root 3 divided by 2. And finally, sine of 11 pi over 6. So we need to use the y value. And sine is negative in quadrant 4. So it's negative 1 half.