Unit circle coordinates for common angles examples

📚 Quick Study Guide: Unit Circle Coordinates

• 🎯 What is the Unit Circle? It's a circle centered at the origin (0,0) with a radius of 1. It's crucial for understanding trigonometric functions because any point $(x, y)$ on the unit circle corresponds to an angle $\theta$ where $x = \cos\theta$ and $y = \sin\theta$.
• ➕ Key Relationships:
  • ↔️ $\cos\theta = x$-coordinate
  • ↕️ $\sin\theta = y$-coordinate
  • 📐 $\tan\theta = \frac{y}{x}$ (when $x \ne 0$)
• 🧠 Common Angles & Their Radians: You'll frequently encounter angles like $0^\circ (0)$, $30^\circ (\frac{\pi}{6})$, $45^\circ (\frac{\pi}{4})$, $60^\circ (\frac{\pi}{3})$, $90^\circ (\frac{\pi}{2})$, $180^\circ (\pi)$, $270^\circ (\frac{3\pi}{2})$, and $360^\circ (2\pi)$. ⏱️
• 📏 Memorizing Quadrant I Coordinates:
  • 🧪 For $30^\circ$ or $\frac{\pi}{6}$: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
  • 🌟 For $45^\circ$ or $\frac{\pi}{4}$: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
  • ✨ For $60^\circ$ or $\frac{\pi}{3}$: $(\frac{1}{2}, \frac{\sqrt{3}}{2})$
  💡 A simple trick: For the numerators of the $x$-coordinates, think $\sqrt{4}, \sqrt{3}, \sqrt{2}, \sqrt{1}, \sqrt{0}$ (for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$) all divided by 2. For $y$-coordinates, reverse it!
• 🔄 Symmetry & Reference Angles: Use the Quadrant I coordinates and the symmetry of the circle to find coordinates in other quadrants.
  • ↖️ Quadrant II ($90^\circ - 180^\circ$): $x$ is negative, $y$ is positive.
  • ↙️ Quadrant III ($180^\circ - 270^\circ$): $x$ is negative, $y$ is negative.
  • ↘️ Quadrant IV ($270^\circ - 360^\circ$): $x$ is positive, $y$ is negative.
• 📋 Essential Unit Circle Coordinates Table: Memorize these fundamental points for quick recall!

  Angle (Degrees)	Angle (Radians)	Coordinates (x, y)
  $0^\circ$	$0$	$(1, 0)$
  $30^\circ$	$\frac{\pi}{6}$	$(\frac{\sqrt{3}}{2}, \frac{1}{2})$
  $45^\circ$	$\frac{\pi}{4}$	$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
  $60^\circ$	$\frac{\pi}{3}$	$(\frac{1}{2}, \frac{\sqrt{3}}{2})$
  $90^\circ$	$\frac{\pi}{2}$	$(0, 1)$
  $120^\circ$	$\frac{2\pi}{3}$	$(-\frac{1}{2}, \frac{\sqrt{3}}{2})$
  $135^\circ$	$\frac{3\pi}{4}$	$(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
  $150^\circ$	$\frac{5\pi}{6}$	$(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
  $180^\circ$	$\pi$	$(-1, 0)$
  $210^\circ$	$\frac{7\pi}{6}$	$(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$
  $225^\circ$	$\frac{5\pi}{4}$	$(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
  $240^\circ$	$\frac{4\pi}{3}$	$(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$
  $270^\circ$	$\frac{3\pi}{2}$	$(0, -1)$
  $300^\circ$	$\frac{5\pi}{3}$	$(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
  $315^\circ$	$\frac{7\pi}{4}$	$(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
  $330^\circ$	$\frac{11\pi}{6}$	$(\frac{\sqrt{3}}{2}, -\frac{1}{2})$
  $360^\circ$	$2\pi$	$(1, 0)$

📝 Practice Quiz

1. What are the coordinates $(x, y)$ for the angle $135^\circ$ on the unit circle?
   1. $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$
   2. $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
   3. $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
   4. $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$
2. Which angle corresponds to the unit circle coordinates $(\frac{1}{2}, \frac{\sqrt{3}}{2})$?
   1. $30^\circ$
   2. $45^\circ$
   3. $60^\circ$
   4. $120^\circ$
3. If an angle $\theta$ has coordinates $(-\frac{\sqrt{3}}{2}, -\frac{1}{2})$ on the unit circle, what is the value of $\theta$ in radians?
   1. $\frac{7\pi}{6}$
   2. $\frac{5\pi}{4}$
   3. $\frac{4\pi}{3}$
   4. $\frac{5\pi}{6}$
4. What is the $x$-coordinate for the angle $\frac{3\pi}{2}$?
   1. $1$
   2. $0$
   3. $-1$
   4. $\frac{\sqrt{2}}{2}$
5. Find the $y$-coordinate for the angle $300^\circ$ on the unit circle.
   1. $\frac{1}{2}$
   2. $-\frac{1}{2}$
   3. $\frac{\sqrt{3}}{2}$
   4. $-\frac{\sqrt{3}}{2}$
6. Which of the following points represents the angle $225^\circ$?
   1. $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
   2. $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
   3. $(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$
   4. $(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
7. If $\sin\theta = -\frac{1}{2}$ and $\cos\theta = \frac{\sqrt{3}}{2}$, what is the angle $\theta$ in degrees?
   1. $150^\circ$
   2. $210^\circ$
   3. $300^\circ$
   4. $330^\circ$