Unit Circle | - Free Printable
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Step-by-step solution for: Unit Circle |
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Show Answer Key & Explanations
Step-by-step solution for: Unit Circle |
To fill in the blank unit circle, we need to determine the coordinates $(x, y)$ for each angle. On a unit circle:
- The x-coordinate is $\cos(\theta)$
- The y-coordinate is $\sin(\theta)$
We will go around the circle starting from $0^\circ$ (or $0$ radians) and moving counter-clockwise. The standard angles are every $30^\circ$ ($\frac{\pi}{6}$) and $45^\circ$ ($\frac{\pi}{4}$).
Here are the key values you need to know:
- $\cos(0) = 1$, $\sin(0) = 0$
- $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$, $\sin(\frac{\pi}{6}) = \frac{1}{2}$
- $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
- $\cos(\frac{\pi}{3}) = \frac{1}{2}$, $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
- $\cos(\frac{\pi}{2}) = 0$, $\sin(\frac{\pi}{2}) = 1$
The signs of the coordinates change depending on the quadrant:
- Quadrant I (top right): $(+, +)$
- Quadrant II (top left): $(-, +)$
- Quadrant III (bottom left): $(-, -)$
- Quadrant IV (bottom right): $(+, -)$
Let's list the points in order, going counter-clockwise from the rightmost point $(1, 0)$.
1. Angle $0$ or $2\pi$:
Coordinate: $(1, 0)$
2. Angle $\frac{\pi}{6}$ ($30^\circ$):
Reference values: $x = \frac{\sqrt{3}}{2}, y = \frac{1}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
3. Angle $\frac{\pi}{4}$ ($45^\circ$):
Reference values: $x = \frac{\sqrt{2}}{2}, y = \frac{\sqrt{2}}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
4. Angle $\frac{\pi}{3}$ ($60^\circ$):
Reference values: $x = \frac{1}{2}, y = \frac{\sqrt{3}}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
5. Angle $\frac{\pi}{2}$ ($90^\circ$):
Coordinate: $(0, 1)$
6. Angle $\frac{2\pi}{3}$ ($120^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
7. Angle $\frac{3\pi}{4}$ ($135^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
8. Angle $\frac{5\pi}{6}$ ($150^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
9. Angle $\pi$ ($180^\circ$):
Coordinate: $(-1, 0)$
10. Angle $\frac{7\pi}{6}$ ($210^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
11. Angle $\frac{5\pi}{4}$ ($225^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
12. Angle $\frac{4\pi}{3}$ ($240^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
13. Angle $\frac{3\pi}{2}$ ($270^\circ$):
Coordinate: $(0, -1)$
14. Angle $\frac{5\pi}{3}$ ($300^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
15. Angle $\frac{7\pi}{4}$ ($315^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
16. Angle $\frac{11\pi}{6}$ ($330^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
---
Final Answer:
Here are the coordinates to fill in the blanks, listed in counter-clockwise order starting from the rightmost point $(1,0)$:
1. $(1, 0)$
2. $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
3. $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
4. $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
5. $(0, 1)$
6. $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
7. $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
8. $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
9. $(-1, 0)$
10. $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
11. $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
12. $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
13. $(0, -1)$
14. $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
15. $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
16. $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
- The x-coordinate is $\cos(\theta)$
- The y-coordinate is $\sin(\theta)$
We will go around the circle starting from $0^\circ$ (or $0$ radians) and moving counter-clockwise. The standard angles are every $30^\circ$ ($\frac{\pi}{6}$) and $45^\circ$ ($\frac{\pi}{4}$).
Here are the key values you need to know:
- $\cos(0) = 1$, $\sin(0) = 0$
- $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$, $\sin(\frac{\pi}{6}) = \frac{1}{2}$
- $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
- $\cos(\frac{\pi}{3}) = \frac{1}{2}$, $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$
- $\cos(\frac{\pi}{2}) = 0$, $\sin(\frac{\pi}{2}) = 1$
The signs of the coordinates change depending on the quadrant:
- Quadrant I (top right): $(+, +)$
- Quadrant II (top left): $(-, +)$
- Quadrant III (bottom left): $(-, -)$
- Quadrant IV (bottom right): $(+, -)$
Let's list the points in order, going counter-clockwise from the rightmost point $(1, 0)$.
1. Angle $0$ or $2\pi$:
Coordinate: $(1, 0)$
2. Angle $\frac{\pi}{6}$ ($30^\circ$):
Reference values: $x = \frac{\sqrt{3}}{2}, y = \frac{1}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
3. Angle $\frac{\pi}{4}$ ($45^\circ$):
Reference values: $x = \frac{\sqrt{2}}{2}, y = \frac{\sqrt{2}}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
4. Angle $\frac{\pi}{3}$ ($60^\circ$):
Reference values: $x = \frac{1}{2}, y = \frac{\sqrt{3}}{2}$
Quadrant I: Both positive.
Coordinate: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
5. Angle $\frac{\pi}{2}$ ($90^\circ$):
Coordinate: $(0, 1)$
6. Angle $\frac{2\pi}{3}$ ($120^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
7. Angle $\frac{3\pi}{4}$ ($135^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
8. Angle $\frac{5\pi}{6}$ ($150^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant II: x is negative, y is positive.
Coordinate: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
9. Angle $\pi$ ($180^\circ$):
Coordinate: $(-1, 0)$
10. Angle $\frac{7\pi}{6}$ ($210^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
11. Angle $\frac{5\pi}{4}$ ($225^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
12. Angle $\frac{4\pi}{3}$ ($240^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant III: Both negative.
Coordinate: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
13. Angle $\frac{3\pi}{2}$ ($270^\circ$):
Coordinate: $(0, -1)$
14. Angle $\frac{5\pi}{3}$ ($300^\circ$):
Reference angle is $\frac{\pi}{3}$. Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
15. Angle $\frac{7\pi}{4}$ ($315^\circ$):
Reference angle is $\frac{\pi}{4}$. Values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
16. Angle $\frac{11\pi}{6}$ ($330^\circ$):
Reference angle is $\frac{\pi}{6}$. Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
Quadrant IV: x is positive, y is negative.
Coordinate: $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
---
Final Answer:
Here are the coordinates to fill in the blanks, listed in counter-clockwise order starting from the rightmost point $(1,0)$:
1. $(1, 0)$
2. $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
3. $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
4. $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
5. $(0, 1)$
6. $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
7. $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
8. $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
9. $(-1, 0)$
10. $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
11. $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
12. $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
13. $(0, -1)$
14. $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
15. $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
16. $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
Parent Tip: Review the logic above to help your child master the concept of blank unit circle worksheet.
