Unit Circle Worksheets - Free Printable
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Step-by-step solution for: Unit Circle Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Unit Circle Worksheets
To solve this problem, we need to find the coordinates $(x, y)$ for each angle on the unit circle.
Key Concepts:
1. Unit Circle: A circle with a radius of $1$ centered at $(0,0)$.
2. Coordinates: For any angle $\theta$, the coordinate is $(\cos \theta, \sin \theta)$.
* The x-coordinate is $\cos \theta$.
* The y-coordinate is $\sin \theta$.
3. Reference Angles: We use special triangles ($30^\circ-60^\circ-90^\circ$ and $45^\circ-45^\circ-90^\circ$) to find the values.
* For $30^\circ$ and $60^\circ$: The numbers involved are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* For $45^\circ$: The numbers involved are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
4. Signs (+ or -): Look at the quadrant (section of the graph) where the angle is.
* Quadrant I (Top Right): Both $x$ and $y$ are positive $(+, +)$.
* Quadrant II (Top Left): $x$ is negative, $y$ is positive $(-, +)$.
* Quadrant III (Bottom Left): Both $x$ and $y$ are negative $(-, -)$.
* Quadrant IV (Bottom Right): $x$ is positive, $y$ is negative $(+, -)$.
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We will go around the circle starting from $0^\circ$ and moving counter-clockwise.
1. Angle $0^\circ$
* Location: Positive x-axis.
* Coordinates: $(1, 0)$
2. Angle $30^\circ$ (Quadrant I)
* Reference angle is $30^\circ$.
* Cosine (x) is $\frac{\sqrt{3}}{2}$, Sine (y) is $\frac{1}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
3. Angle $45^\circ$ (Quadrant I)
* Reference angle is $45^\circ$.
* Cosine and Sine are both $\frac{\sqrt{2}}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
4. Angle $60^\circ$ (Quadrant I)
* Reference angle is $60^\circ$.
* Cosine (x) is $\frac{1}{2}$, Sine (y) is $\frac{\sqrt{3}}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
5. Angle $90^\circ$
* Location: Positive y-axis.
* Coordinates: $(0, 1)$
6. Angle $120^\circ$ (Quadrant II)
* Reference angle is $60^\circ$ ($180 - 120 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(-, +)$ because it's in Quadrant II.
* Answer: $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
7. Angle $135^\circ$ (Quadrant II)
* Reference angle is $45^\circ$ ($180 - 135 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(-, +)$
* Answer: $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
8. Angle $150^\circ$ (Quadrant II)
* Reference angle is $30^\circ$ ($180 - 150 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(-, +)$
* Answer: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
9. Angle $180^\circ$
* Location: Negative x-axis.
* Coordinates: $(-1, 0)$
10. Angle $210^\circ$ (Quadrant III)
* Reference angle is $30^\circ$ ($210 - 180 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(-, -)$ because it's in Quadrant III.
* Answer: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
11. Angle $225^\circ$ (Quadrant III)
* Reference angle is $45^\circ$ ($225 - 180 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(-, -)$
* Answer: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
12. Angle $240^\circ$ (Quadrant III)
* Reference angle is $60^\circ$ ($240 - 180 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(-, -)$
* Answer: $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
13. Angle $270^\circ$
* Location: Negative y-axis.
* Coordinates: $(0, -1)$
14. Angle $300^\circ$ (Quadrant IV)
* Reference angle is $60^\circ$ ($360 - 300 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(+, -)$ because it's in Quadrant IV.
* Answer: $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
15. Angle $315^\circ$ (Quadrant IV)
* Reference angle is $45^\circ$ ($360 - 315 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(+, -)$
* Answer: $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
16. Angle $330^\circ$ (Quadrant IV)
* Reference angle is $30^\circ$ ($360 - 330 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(+, -)$
* Answer: $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
---
Final Answer:
Here are the coordinates to fill in the blanks, going counter-clockwise from the right side ($0^\circ$):
* $0^\circ$: $(1, 0)$
* $30^\circ$: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
* $45^\circ$: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
* $60^\circ$: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
* $90^\circ$: $(0, 1)$
* $120^\circ$: $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
* $135^\circ$: $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
* $150^\circ$: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
* $180^\circ$: $(-1, 0)$
* $210^\circ$: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
* $225^\circ$: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
* $240^\circ$: $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
* $270^\circ$: $(0, -1)$
* $300^\circ$: $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
* $315^\circ$: $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
* $330^\circ$: $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
Key Concepts:
1. Unit Circle: A circle with a radius of $1$ centered at $(0,0)$.
2. Coordinates: For any angle $\theta$, the coordinate is $(\cos \theta, \sin \theta)$.
* The x-coordinate is $\cos \theta$.
* The y-coordinate is $\sin \theta$.
3. Reference Angles: We use special triangles ($30^\circ-60^\circ-90^\circ$ and $45^\circ-45^\circ-90^\circ$) to find the values.
* For $30^\circ$ and $60^\circ$: The numbers involved are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* For $45^\circ$: The numbers involved are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$.
4. Signs (+ or -): Look at the quadrant (section of the graph) where the angle is.
* Quadrant I (Top Right): Both $x$ and $y$ are positive $(+, +)$.
* Quadrant II (Top Left): $x$ is negative, $y$ is positive $(-, +)$.
* Quadrant III (Bottom Left): Both $x$ and $y$ are negative $(-, -)$.
* Quadrant IV (Bottom Right): $x$ is positive, $y$ is negative $(+, -)$.
---
Step-by-Step Solution
We will go around the circle starting from $0^\circ$ and moving counter-clockwise.
1. Angle $0^\circ$
* Location: Positive x-axis.
* Coordinates: $(1, 0)$
2. Angle $30^\circ$ (Quadrant I)
* Reference angle is $30^\circ$.
* Cosine (x) is $\frac{\sqrt{3}}{2}$, Sine (y) is $\frac{1}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
3. Angle $45^\circ$ (Quadrant I)
* Reference angle is $45^\circ$.
* Cosine and Sine are both $\frac{\sqrt{2}}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
4. Angle $60^\circ$ (Quadrant I)
* Reference angle is $60^\circ$.
* Cosine (x) is $\frac{1}{2}$, Sine (y) is $\frac{\sqrt{3}}{2}$.
* Signs: $(+, +)$
* Answer: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
5. Angle $90^\circ$
* Location: Positive y-axis.
* Coordinates: $(0, 1)$
6. Angle $120^\circ$ (Quadrant II)
* Reference angle is $60^\circ$ ($180 - 120 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(-, +)$ because it's in Quadrant II.
* Answer: $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
7. Angle $135^\circ$ (Quadrant II)
* Reference angle is $45^\circ$ ($180 - 135 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(-, +)$
* Answer: $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
8. Angle $150^\circ$ (Quadrant II)
* Reference angle is $30^\circ$ ($180 - 150 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(-, +)$
* Answer: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
9. Angle $180^\circ$
* Location: Negative x-axis.
* Coordinates: $(-1, 0)$
10. Angle $210^\circ$ (Quadrant III)
* Reference angle is $30^\circ$ ($210 - 180 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(-, -)$ because it's in Quadrant III.
* Answer: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
11. Angle $225^\circ$ (Quadrant III)
* Reference angle is $45^\circ$ ($225 - 180 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(-, -)$
* Answer: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
12. Angle $240^\circ$ (Quadrant III)
* Reference angle is $60^\circ$ ($240 - 180 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(-, -)$
* Answer: $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
13. Angle $270^\circ$
* Location: Negative y-axis.
* Coordinates: $(0, -1)$
14. Angle $300^\circ$ (Quadrant IV)
* Reference angle is $60^\circ$ ($360 - 300 = 60$).
* Values are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.
* Signs: $(+, -)$ because it's in Quadrant IV.
* Answer: $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
15. Angle $315^\circ$ (Quadrant IV)
* Reference angle is $45^\circ$ ($360 - 315 = 45$).
* Values are $\frac{\sqrt{2}}{2}$.
* Signs: $(+, -)$
* Answer: $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
16. Angle $330^\circ$ (Quadrant IV)
* Reference angle is $30^\circ$ ($360 - 330 = 30$).
* Values are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$.
* Signs: $(+, -)$
* Answer: $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
---
Final Answer:
Here are the coordinates to fill in the blanks, going counter-clockwise from the right side ($0^\circ$):
* $0^\circ$: $(1, 0)$
* $30^\circ$: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
* $45^\circ$: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
* $60^\circ$: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
* $90^\circ$: $(0, 1)$
* $120^\circ$: $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
* $135^\circ$: $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
* $150^\circ$: $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
* $180^\circ$: $(-1, 0)$
* $210^\circ$: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
* $225^\circ$: $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
* $240^\circ$: $\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
* $270^\circ$: $(0, -1)$
* $300^\circ$: $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$
* $315^\circ$: $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
* $330^\circ$: $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
Parent Tip: Review the logic above to help your child master the concept of unit circle trig worksheet.
