The unit circle is called the "unit" circle because it has a radius of 1.
Question 1
1.
What is the length of the radius of the unit circle?
Question 2
2.
When the unit circle is drawn over an x- and y-axis it has some important coordinate points (x,y) along the edge of the circle.
What is the coordinate point at 90\degree?
Hint: Look at the image above.
Type your answer as an ordered pair (x,y).
Question 3
3.
What is the coordinate point at 180° ?
Type your answer as an ordered pair (x,y).
Question 4
4.
What is the coordinate point at 270° ?
Type your answer as an ordered pair (x,y).
Question 5
5.
What is the coordinate point at 2\pi radians ?
Type your answer as an ordered pair (x,y).
Question 6
6.
What's the coordinate point at 30\degree? __________
To find the coordinate point at 30° we can create a right triangle with the 30° reference angle.
Keep scrolling to see how!
Question 7
7.
We're going to be drawing triangles inside the unit circle, using the special angles of the unit circle.
These are the special angles of the unit circle:
Where else have you seen these angles?
Question 8
8.
Let's take a closer look at ONLY the 30\degree angle in Quadrant I.
If we draw a vertical line from the edge of the circle down to the x-axis it creates a right triangle:
What is the measure of the missing angle at the top of the triangle?
Question 9
9.
If the radius of the unit circle is 1, what is the length of the hypotenuse of this triangle?
Now we need to find the two missing side lengths x and y using the 30-60-90 special right triangle ratios.
You cannot use sin or cos because we need exact fractions, no decimals!
Recall how to set up the table for 30-60-90
Question 10
10.
What is the exact value of x?
No decimal answers so you must use the table above!
Question 11
11.
What is the exact value of y?
No decimal answers so you must use the table above!
Question 12
12.
The x-value and y-value you just found make up the coordinate point (x,y) for the 30\degree angle on the unit circle.
Notice how the __________ of the triangle represents the x-value of the point and the __________ of the triangle represents the y-value of the point on the circle.
Since the __________ of the triangle is the radius of the circle, that will always be __________
Question 13
13.
Looks like we found 3 of the 5 coordinate points in Quadrant 1.
Let's find the other two! Which reference angle does the middle point in Q1 represent?
Question 14
14.
If we create a right triangle with the 45\degree reference angle, what is the measure of the missing angle at the top of the triangle?
Question 15
15.
Use the 45-45-90 table to find exact value of x.
No decimals and remember to rationalize the denominator!
Question 16
16.
Use the same table to find exact value of y.
No decimals and remember to rationalize the denominator!
Question 17
17.
The x-value and y-value you just found make up the coordinate point (x,y) for the 45\degree angle on the unit circle.
Notice how the __________ of the triangle represents the x-value of the point and the __________ of the triangle represents the y-value of the point on the circle.
Since the __________ of the triangle is the radius of the circle, that will always be __________
Question 18
18.
We have ONE more coordinate point to find in Quadrant 1.
Which reference angle does the last missing point represent?
Question 19
19.
If we create a right triangle with the 60\degree reference angle, what is the measure of the missing angle at the top of the triangle?
Question 20
20.
Notice that we have another 30-60-90 triangle, except the 30\degree and 60\degree angles switched places. How do you think that will effect the base (x) and height (y) of the triangle?
Question 21
21.
If you were to find x and y using the 30-60-90 table,
you would find that the values do in fact __________ from the 30\degree reference angle
Question 22
22.
List the values in order from least to greatest.
Question 23
23.
Start at 0\degree and rotate up counter-clockwise.
As the angles in the unit circle increase:
- the x-value starts at 1 and gets __________ until x=0 at 90\degree. This is because the __________ of the right triangles we drew got smaller and smaller.
- the y-value starts a 0 and gets __________ until y=__________ at 90°. This is because the __________ of the right triangles we drew got taller and taller.
On the unit circle on your Radians worksheet, write in the coordinate points for Quadrants 1 and 2.
Notice how the reference angle (denominator) determines the coordinate points.
For example, the \frac{\pi }{3} reference angles in Q1 and Q2 have similar values.
Question 24
24.
Notice how the coordinate points in Quadrant 2 are the same as Quadrant 1, except now all of the __________ are negative.
Question 25
25.
The coordinate point for \frac{\pi }{6} in Q1 is
What is the coordinate point for \frac{5\pi }{6} in Q2?
Question 26
26.
When you list out the coordinate points in Quadrant III, what do you think will happen to the signs of the values?
Question 27
27.
The coordinate point for \frac{\pi }{4} in Q1 is
What is the coordinate point for \frac{5\pi }{4} in Q3?
Question 28
28.
The coordinate point for \frac{\pi }{4} in Q1 is
What is the coordinate point for \frac{7\pi }{4} in Q4?
Hint: think about if the x-value or y-value would be negative in Quadrant IV
Question 29
29.
Match the correct signs of each coordinate point (x,y) for each Quadrant.
(-,+)
(-,-)
(+,-)
(+,+)
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
Question 30
30.
The coordinate point for \frac{\pi }{3} in Q1 is
What is the coordinate point for \frac{5\pi }{3} in Q4?
Fill in the rest of the coordinate points all around the unit circle on your worksheet. Raise your hand when you're done so I can check your work!