AP CALCULUS The Unit circle Review ppt download - Free Printable

To fill in the unit circle, we need to determine two things for each point:
1. The Angle: Measured in degrees (°) and radians. We start at 0° on the right and go counter-clockwise. The circle is divided into 12 equal slices, so each slice is $360^\circ / 12 = 30^\circ$. In radians, a full circle is $2\pi$, so each slice is $2\pi / 12 = \pi/6$.
2. The Coordinates $(x, y)$: On a unit circle, the coordinates correspond to $(\cos \theta, \sin \theta)$.
* x-coordinate: $\cos(\text{angle})$
* y-coordinate: $\sin(\text{angle})$

Here are the key values you will see repeated:
* For $30^\circ$ ($\frac{\pi}{6}$) angles: The numbers involved are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$. Since $30^\circ$ is close to the x-axis, the x-value is larger: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$.
* For $45^\circ$ ($\frac{\pi}{4}$) angles: The numbers are equal: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.
* For $60^\circ$ ($\frac{\pi}{3}$) angles: The numbers involved are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$. Since $60^\circ$ is closer to the y-axis, the y-value is larger: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

Signs by Quadrant:
* Quadrant I (Top Right): Both positive $(+, +)$
* Quadrant II (Top Left): x is negative, y is positive $(-, +)$
* Quadrant III (Bottom Left): Both negative $(-, -)$
* Quadrant IV (Bottom Right): x is positive, y is negative $(+, -)$

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Step-by-Step Solution

We will list the values starting from the right (0°) and moving counter-clockwise around the circle.

1. Point at 0° (Rightmost)
* Angle: $0^\circ$, $0$ rad
* Coordinates: $(1, 0)$
* Positive/Negative: Positive: $x$, Negative: None (or y)

2. Point at 30° (First tick up)
* Angle: $30^\circ$, $\frac{\pi}{6}$
* Coordinates: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
* Positive/Negative: Positive: $x, y$; Negative: None

3. Point at 45° (Second tick up)
* Angle: $45^\circ$, $\frac{\pi}{4}$
* Coordinates: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
* Positive/Negative: Positive: $x, y$; Negative: None

4. Point at 60° (Third tick up)
* Angle: $60^\circ$, $\frac{\pi}{3}$
* Coordinates: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
* Positive/Negative: Positive: $x, y$; Negative: None

5. Point at 90° (Top)
* Angle: $90^\circ$, $\frac{\pi}{2}$
* Coordinates: $(0, 1)$
* Positive/Negative: Positive: $y$, Negative: None (or x)

6. Point at 120° (Top Left, first tick)
* Angle: $120^\circ$, $\frac{2\pi}{3}$
* Coordinates: $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
* Positive/Negative: Positive: $y$; Negative: $x$

7. Point at 135° (Top Left, second tick)
* Angle: $135^\circ$, $\frac{3\pi}{4}$
* Coordinates: $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
* Positive/Negative: Positive: $y$; Negative: $x$

8. Point at 150° (Top Left, third tick)
* Angle: $150^\circ$, $\frac{5\pi}{6}$
* Coordinates: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
* Positive/Negative: Positive: $y$; Negative: $x$

9. Point at 180° (Leftmost)
* Angle: $180^\circ$, $\pi$
* Coordinates: $(-1, 0)$
* Positive/Negative: Positive: None (or y); Negative: $x$

10. Point at 210° (Bottom Left, first tick)
* Angle: $210^\circ$, $\frac{7\pi}{6}$
* Coordinates: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
* Positive/Negative: Positive: None; Negative: $x, y$

11. Point at 225° (Bottom Left, second tick)
* Angle: $225^\circ$, $\frac{5\pi}{4}$
* Coordinates: $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
* Positive/Negative: Positive: None; Negative: $x, y$

12. Point at 240° (Bottom Left, third tick)
* Angle: $240^\circ$, $\frac{4\pi}{3}$
* Coordinates: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
* Positive/Negative: Positive: None; Negative: $x, y$

13. Point at 270° (Bottom)
* Angle: $270^\circ$, $\frac{3\pi}{2}$
* Coordinates: $(0, -1)$
* Positive/Negative: Positive: None (or x); Negative: $y$

14. Point at 300° (Bottom Right, first tick)
* Angle: $300^\circ$, $\frac{5\pi}{3}$
* Coordinates: $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
* Positive/Negative: Positive: $x$; Negative: $y$

15. Point at 315° (Bottom Right, second tick)
* Angle: $315^\circ$, $\frac{7\pi}{4}$
* Coordinates: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
* Positive/Negative: Positive: $x$; Negative: $y$

16. Point at 330° (Bottom Right, third tick)
* Angle: $330^\circ$, $\frac{11\pi}{6}$
* Coordinates: $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
* Positive/Negative: Positive: $x$; Negative: $y$

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Final Answer:

Here is the data to fill into the blanks on your worksheet, organized by position on the circle.

Axis Points (The Crosshairs):
* Right ($0^\circ$): Angle: $0^\circ, 0$ | Coords: $(1, 0)$ | Pos: $x$ | Neg: $y$
* Top ($90^\circ$): Angle: $90^\circ, \frac{\pi}{2}$ | Coords: $(0, 1)$ | Pos: $y$ | Neg: $x$
* Left ($180^\circ$): Angle: $180^\circ, \pi$ | Coords: $(-1, 0)$ | Pos: $y$ | Neg: $x$
* Bottom ($270^\circ$): Angle: $270^\circ, \frac{3\pi}{2}$ | Coords: $(0, -1)$ | Pos: $x$ | Neg: $y$

Quadrant I (Top Right) - Going Up:
1. $30^\circ$: Angle: $30^\circ, \frac{\pi}{6}$ | Coords: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
2. $45^\circ$: Angle: $45^\circ, \frac{\pi}{4}$ | Coords: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
3. $60^\circ$: Angle: $60^\circ, \frac{\pi}{3}$ | Coords: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$

Quadrant II (Top Left) - Going Left:
4. $120^\circ$: Angle: $120^\circ, \frac{2\pi}{3}$ | Coords: $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
5. $135^\circ$: Angle: $135^\circ, \frac{3\pi}{4}$ | Coords: $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
6. $150^\circ$: Angle: $150^\circ, \frac{5\pi}{6}$ | Coords: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$

Quadrant III (Bottom Left) - Going Down:
7. $210^\circ$: Angle: $210^\circ, \frac{7\pi}{6}$ | Coords: $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
8. $225^\circ$: Angle: $225^\circ, \frac{5\pi}{4}$ | Coords: $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
9. $240^\circ$: Angle: $240^\circ, \frac{4\pi}{3}$ | Coords: $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$

Quadrant IV (Bottom Right) - Going Right:
10. $300^\circ$: Angle: $300^\circ, \frac{5\pi}{3}$ | Coords: $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
11. $315^\circ$: Angle: $315^\circ, \frac{7\pi}{4}$ | Coords: $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
12. $330^\circ$: Angle: $330^\circ, \frac{11\pi}{6}$ | Coords: $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$