Unit Circle Quiz - Free Printable

To fill in the unit circle correctly, we need to find two things for each point:
1. The Angle (in parentheses): This is usually measured in radians. We start at 0 on the right and go counter-clockwise. The circle is divided into 12 equal sections, so each step adds $\frac{\pi}{6}$ (or $30^\circ$).
2. The Coordinates $(x, y)$: These are the exact values for $(\cos \theta, \sin \theta)$.
* x-value: Look at how far left or right the point is.
* y-value: Look at how far up or down the point is.

Here is the step-by-step solution going around the circle starting from the right (3 o'clock position) and moving counter-clockwise.

Step 1: The "Easy" Points (Axes)

These points lie directly on the horizontal or vertical lines. Their coordinates are always $0$, $1$, or $-1$.

* Right (0 radians): Angle is $0$. Point is $(1, 0)$.
* Top ($\frac{\pi}{2}$ radians): Angle is $\frac{\pi}{2}$. Point is $(0, 1)$.
* Left ($\pi$ radians): Angle is $\pi$. Point is $(-1, 0)$.
* Bottom ($\frac{3\pi}{2}$ radians): Angle is $\frac{3\pi}{2}$. Point is $(0, -1)$.

Step 2: The First Quadrant (Top Right)

In this section, both x and y are positive. We use denominators of 2.
* $30^\circ$ mark: Angle is $\frac{\pi}{6}$.
* x is further out than y. So, x is $\frac{\sqrt{3}}{2}$ and y is $\frac{1}{2}$.
* Point: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
* $45^\circ$ mark: Angle is $\frac{\pi}{4}$.
* x and y are equal.
* Point: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
* $60^\circ$ mark: Angle is $\frac{\pi}{3}$.
* y is further out than x. So, y is $\frac{\sqrt{3}}{2}$ and x is $\frac{1}{2}$.
* Point: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$

Step 3: The Second Quadrant (Top Left)

In this section, x is negative (left) and y is positive (up).
* $120^\circ$ mark: Angle is $\frac{2\pi}{3}$.
* Mirror of the $60^\circ$ point. x becomes negative.
* Point: $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
* $135^\circ$ mark: Angle is $\frac{3\pi}{4}$.
* Mirror of the $45^\circ$ point. x becomes negative.
* Point: $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
* $150^\circ$ mark: Angle is $\frac{5\pi}{6}$.
* Mirror of the $30^\circ$ point. x becomes negative.
* Point: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$

Step 4: The Third Quadrant (Bottom Left)

In this section, both x and y are negative.
* $210^\circ$ mark: Angle is $\frac{7\pi}{6}$.
* Mirror of the $30^\circ$ point. Both become negative.
* Point: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
* $225^\circ$ mark: Angle is $\frac{5\pi}{4}$.
* Mirror of the $45^\circ$ point. Both become negative.
* Point: $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
* $240^\circ$ mark: Angle is $\frac{4\pi}{3}$.
* Mirror of the $60^\circ$ point. Both become negative.
* Point: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$

Step 5: The Fourth Quadrant (Bottom Right)

In this section, x is positive (right) and y is negative (down).
* $300^\circ$ mark: Angle is $\frac{5\pi}{3}$.
* Mirror of the $60^\circ$ point. Only y becomes negative.
* Point: $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
* $315^\circ$ mark: Angle is $\frac{7\pi}{4}$.
* Mirror of the $45^\circ$ point. Only y becomes negative.
* Point: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
* $330^\circ$ mark: Angle is $\frac{11\pi}{6}$.
* Mirror of the $30^\circ$ point. Only y becomes negative.
* Point: $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$

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Final Answer:

Here are the values to fill in on the diagram, listed in order starting from the right (3 o'clock) and going counter-clockwise around the circle.

Format: ( Angle ) ... [ x-coordinate , y-coordinate ]

1. ( $0$ ) ... [ $1$ , $0$ ]
2. ( $\frac{\pi}{6}$ ) ... [ $\frac{\sqrt{3}}{2}$ , $\frac{1}{2}$ ]
3. ( $\frac{\pi}{4}$ ) ... [ $\frac{\sqrt{2}}{2}$ , $\frac{\sqrt{2}}{2}$ ]
4. ( $\frac{\pi}{3}$ ) ... [ $\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ ]
5. ( $\frac{\pi}{2}$ ) ... [ $0$ , $1$ ]
6. ( $\frac{2\pi}{3}$ ) ... [ $-\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ ]
7. ( $\frac{3\pi}{4}$ ) ... [ $-\frac{\sqrt{2}}{2}$ , $\frac{\sqrt{2}}{2}$ ]
8. ( $\frac{5\pi}{6}$ ) ... [ $-\frac{\sqrt{3}}{2}$ , $\frac{1}{2}$ ]
9. ( $\pi$ ) ... [ $-1$ , $0$ ]
10. ( $\frac{7\pi}{6}$ ) ... [ $-\frac{\sqrt{3}}{2}$ , $-\frac{1}{2}$ ]
11. ( $\frac{5\pi}{4}$ ) ... [ $-\frac{\sqrt{2}}{2}$ , $-\frac{\sqrt{2}}{2}$ ]
12. ( $\frac{4\pi}{3}$ ) ... [ $-\frac{1}{2}$ , $-\frac{\sqrt{3}}{2}$ ]
13. ( $\frac{3\pi}{2}$ ) ... [ $0$ , $-1$ ]
14. ( $\frac{5\pi}{3}$ ) ... [ $\frac{1}{2}$ , $-\frac{\sqrt{3}}{2}$ ]
15. ( $\frac{7\pi}{4}$ ) ... [ $\frac{\sqrt{2}}{2}$ , $-\frac{\sqrt{2}}{2}$ ]
16. ( $\frac{11\pi}{6}$ ) ... [ $\frac{\sqrt{3}}{2}$ , $-\frac{1}{2}$ ]