Introduction to the Unit Circle

Show your work for each problem.

1.
What is the radius of the unit circle?
1. a)2
2. b)0
3. c)1
4. d)
Answer: 0
The unit circle has a radius of exactly 1. That's why it's called the 'unit' circle - the radius is one unit.
2.
What are the coordinates of the center of the unit circle?
1. a)
2. b)
3. c)
4. d)
Answer:
The unit circle is centered at the origin . The other options are points ON the unit circle, not its center.
3.
What is ?
Answer: 1
At , the point on the unit circle is at . Since cosine is the x-coordinate, .
4.
What is ?
Answer: 1
At , the point on the unit circle is at . Since sine is the y-coordinate, .
5.
On the unit circle, what does the x-coordinate of a point represent?
1. a)
2. b)
3. c)The radius
4. d)
Answer:
On the unit circle, the x-coordinate represents and the y-coordinate represents . The coordinates are .
6.
What is ?
Answer: -1
At , the point on the unit circle is at . Since cosine is the x-coordinate, .
7.
What is ?
Answer: -1
At , the point on the unit circle is at . Since sine is the y-coordinate, .
8.
In which quadrant are both sine and cosine positive?
1. a)Quadrant III
2. b)Quadrant II
3. c)Quadrant IV
4. d)Quadrant I
Answer: Quadrant I
In Quadrant I (upper right), both x and y are positive. Since and , both are positive in Quadrant I.
9.
Find using the unit circle.
Answer: sqrt(2)/2
1. At 45 degrees, what is special about the x and y coordinates? equal
2. If x = y and x^2 + y^2 = 1, what is x^2? 1/2
3. What is the value of x (and therefore sin(45))? sqrt(2)/2
10.
Find the coordinates of the point on the unit circle at .
Answer: (-1, 0)
1. Starting from the positive x-axis, how many degrees do you rotate to reach 180 degrees? 180
2. Where does this put you on the circle? left
3. What is the x-coordinate (cosine) at this point? -1
4. What is the y-coordinate (sine) at this point? 0
11.
In which quadrant is positive and negative?
1. a)Quadrant II
2. b)Quadrant III
3. c)Quadrant IV
4. d)Quadrant I
Answer: Quadrant II
In Quadrant II (upper left), x is negative and y is positive. Since and , sine is positive and cosine is negative in Quadrant II.
12.
What is ?
Answer: 0
At , we've completed a full rotation and returned to the starting point . Since sine is the y-coordinate, . Notice this is the same as !
13.
Find using reference angles.
Answer: -1/2
1. What quadrant is 120 degrees in? II
2. What is the reference angle? (180 - 120) 60
3. What is cos(60)? 1/2
4. In Quadrant II, is cosine positive or negative? negative
5. So what is cos(120)? -1/2
14.
If a point on the unit circle has coordinates , verify it lies on the unit circle.
Answer: 1
1. What is the equation of the unit circle? x^2 + y^2 = 1
2. What is x^2 when x = 0.6? 0.36
3. What is y^2 when y = 0.8? 0.64
4. What is x^2 + y^2? 1
15.
What is true about for any angle ?
1. a)It always equals 2
2. b)It depends on the angle
3. c)It always equals 0
4. d)It always equals 1
Answer: It always equals 1
Since is always on the unit circle, it must satisfy . Therefore for ALL angles. This is the Pythagorean identity!