The Unit Circle
What is the unit circle?
The unit circle is a circle with radius 1 that is representative of trigonometric values of the cosine and sine functions expressed in radians. While any value could be found on the unit circle, we will focus on values of common angles around the circle.
How is the unit circle made?
The unit circle is created so that the circle always has a radius of 1 and is centered at the origin of the coordinate plane. To find the various values of points on the unit circle, the two special right triangles are placed in the circle with one endpoint of the hypotenuse touching the origin and the other endpoint of the hypotenuse touching the circle.
Notice that since the hypotenuse touches the origin (the center of the unit circle) and the circle itself, that makes the hypotenuse of each special right triangle the radius of the circle. Therefore the length of the hypotenuse of each special right triangle is 1 (when we are dealing with the unit circle).
The unit circle is a circle with radius 1 that is representative of trigonometric values of the cosine and sine functions expressed in radians. While any value could be found on the unit circle, we will focus on values of common angles around the circle.
How is the unit circle made?
The unit circle is created so that the circle always has a radius of 1 and is centered at the origin of the coordinate plane. To find the various values of points on the unit circle, the two special right triangles are placed in the circle with one endpoint of the hypotenuse touching the origin and the other endpoint of the hypotenuse touching the circle.
Notice that since the hypotenuse touches the origin (the center of the unit circle) and the circle itself, that makes the hypotenuse of each special right triangle the radius of the circle. Therefore the length of the hypotenuse of each special right triangle is 1 (when we are dealing with the unit circle).
The Special Right Triangles
Special Right Triangles in Radians with Hypotenuse (Radius) of 1 Putting Special Right Triangles into the Unit Circle |
Everything we do in calculus will be in radian measure. Remember, to go from degrees to radians, multiply by the conversion π radians / 180°.
Notice that the point on the unit circle, (x, y), corresponds with a width and height of a triangle. Therefore, the measurements of the sides of the special right triangle become the coordinates of the point on the unit circle.
We talk about angles of the unit circle always starting from the positive x-axis and rotating counterclockwise around the circle as shown with the green arrow above. So when looking at the angle π/4, the corresponding unit circle point is (√2/2, √2/2). Also notice that a point on the unit circle is not only (x, y), but also (cosθ, sinθ), where θ is the measure of the angle in radians. For the picture above, this tells us that (cos(π/4), sin(π/4)) = (√2/2, √2/2). So we can deduce that cos(π/4) = √2/2 and that sin(π/4) = √2/2. This process of placing special right triangles into quadrants of the unit circle can be repeated over and over to find coordinates for a 30°, 45°, and 60° angle in each quadrant. The signs of the x- and y-coordinates will correspond with the location of the points in the quadrants. So for Quadrant II, all cosine values will be negative and all sine values will be positive. In Quadrant III, all cosine and sine values will be negative. In Quadrant IV, all cosine values will be positive, while sine values will be negative. |
That Seems Like a Lot of Work...
I agree. It would be a lot of work if you had to consider the triangles and derive the circle every time you needed to know the sine or cosine value of some angle. So let's make this a little easier on you!
What you need to know:
✓ The angles of the unit circle in radians
✓ Equivalent angles to those of the unit circle
✓ The corresponding coordinates (points) that go with each angle
✓ The cosine and sine value of EVERY angle in the unit circle
Note: You should be able to recite these at random. For example, if you come to class on the first day of school and I say, "Pop Quiz! Tell me the cos(5π/4) and sin(3π/2)," you should be able to do so in a heartbeat! You will need to know these values throughout the year in the work we do and on the AP Test in May.
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Equivalent Radian Measures of the Unit Circle
As we "travel" around the unit circle, the radian measures will continue to grow even though the circle only shows 0 through 11π/6. For example, observe that if we travel from our starting point, 0 (the positive x-axis), and travel to π, we have traveled around half of the circle. If we travel around another half of the circle, we have traveled π more in distance. So we would think we would be at the location 2π, but the circle shows that location as 0. What you can get from this is that 0 and 2π are essentially the same thing with regard to the unit circle and will have the same values. So what would an equivalent value be for 7π/2? If you said 3π/2, you are correct!
If you didn't get that, here's how it works:
1. Start at 0.
2. Notice that our denominator is 2. So I want to travel to all the places with a 2 in the denominator.
3. All I need to do is count around the circle until I get to 7.
4. The first location with a 2 in the denominator is π/2, a quarter of the circle away from 0. This tells me I'm going to travel in quarters.
I also know that π/2 is the same as 1π/2.
5. Start counting around the circle by traveling in quarter circles. The next spot would be 2π/2 on the negative side of the x-axis.
We know, though, that 2π/2 simplifies to π. So we are doing things right.
6. Keep counting and tracing around the circle: 3π/2, 4π/2, 5π/2, 6π/2, 7π/2. You should end at the same location as 3π/2.
Some people find it easier to memorize the unit circle values using the chart below. During quizzes (and pop quizzes) in our class throughout the year, I will use the table format to test your knowledge. However, you will still need to know what the circle looks like in order to calculate equivalent radian measurements.
Tangent does appear in this chart. While you are not required to memorize the tangent values, you will need to be able to calculate them. Recall that tangent = sine / cosine. So, since you will know your cosine and sine values from your unit circle points, all you have to do is divide! (x, y) = (cosθ, sinθ) tanθ = y / x tanθ = sinθ / cosθ Your in-class quizzes will only consist of the first three columns of the chart to the left. |
Assignments
Quiz
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Practice: First, spend some time studying the unit circle and the information presented above. Work on memorizing the unit circle values and finding equivalent radian measures of the unit circle.
Document: Tricks for memorizing the unit circle values. Document: Blank unit circle wheel and chart. Document: Practice assignment with answers. Assignment: The assignment for this unit is below. Remember, this is not due by any date...ever. It's just extra practice for you to become more familiar with the material! Once you have completed and submitted the assignment above, you may take the quiz below. The quiz is not self grading, but will give you an idea of what types of questions I might ask about the unit circle in the future. You do not need to submit the quiz. |