The Unit Circle: Complete Guide
Introduction to the Unit Circle
The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. It is one of the most important tools in trigonometry, serving as a visual representation of trigonometric functions and their relationships.
Key features of the unit circle:
- Radius = 1 unit
- Center at the origin (0,0)
- Any point (x,y) on the unit circle corresponds to (cos θ, sin θ) where θ is the angle from the positive x-axis
- The equation of the unit circle is x² + y² = 1
Interactive Unit Circle
For angle 0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
Unit Circle Values
| Angle (Degrees) | Angle (Radians) | sin θ | cos θ | tan θ | Point (x,y) |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 | (√3/2, 1/2) |
| 45° | π/4 | 1/√2 | 1/√2 | 1 | (1/√2, 1/√2) |
| 60° | π/3 | √3/2 | 1/2 | √3 | (1/2, √3/2) |
| 90° | π/2 | 1 | 0 | undefined | (0, 1) |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 | (-1/2, √3/2) |
| 180° | π | 0 | -1 | 0 | (-1, 0) |
| 270° | 3π/2 | -1 | 0 | undefined | (0, -1) |
| 360° | 2π | 0 | 1 | 0 | (1, 0) |
Degrees and Radians
Angles on the unit circle can be measured in both degrees and radians. Here's how they relate:
Conversion Formulas:
To convert from degrees to radians:
radians = degrees × (π / 180)
To convert from radians to degrees:
degrees = radians × (180 / π)
Key Radian Values:
- Full circle: 2π radians (360°)
- Half circle: π radians (180°)
- Quarter circle: π/2 radians (90°)
- Sixth of a circle: π/3 radians (60°)
- Eighth of a circle: π/4 radians (45°)
- Twelfth of a circle: π/6 radians (30°)
Degree-Radian Converter
Problem-Solving with the Unit Circle
Method 1: Finding Trigonometric Values Using the Unit Circle
Example 1: Find sin(60°) and cos(60°)
Solution:
- Locate 60° (π/3 radians) on the unit circle
- The coordinates of this point are (1/2, √3/2)
- Since any point on the unit circle is (cos θ, sin θ), we have:
- cos(60°) = 1/2
- sin(60°) = √3/2
Method 2: Reference Angles
A reference angle is the acute angle (angle less than 90°) formed by the terminal side of the angle and the x-axis.
Example 2: Find cos(150°)
Solution:
- The reference angle for 150° is 180° - 150° = 30°
- We know that cos(30°) = √3/2
- Since 150° is in the second quadrant, where cosine is negative:
- cos(150°) = -cos(30°) = -√3/2
Method 3: ASTC Rule for Signs
Remember which functions are positive in which quadrants using the mnemonic "ASTC" (All, Sine, Tangent, Cosine):
- Quadrant I (0° to 90°): All functions are positive (A)
- Quadrant II (90° to 180°): Only Sine is positive (S)
- Quadrant III (180° to 270°): Only Tangent is positive (T)
- Quadrant IV (270° to 360°): Only Cosine is positive (C)
Example 3: Find sin(300°)
Solution:
- The reference angle for 300° is 360° - 300° = 60°
- We know that sin(60°) = √3/2
- Since 300° is in the fourth quadrant, where sine is negative:
- sin(300°) = -sin(60°) = -√3/2
Method 4: Converting Between Radians and Degrees
Example 4: Convert 5π/4 radians to degrees
Solution:
- Use the conversion formula: degrees = radians × (180/π)
- degrees = 5π/4 × (180/π) = 5 × 180/4 = 900/4 = 225°
- Therefore, 5π/4 radians = 225°
Method 5: Finding All Solutions to Trigonometric Equations
Example 5: Find all solutions to sin(θ) = 1/2 for 0° ≤ θ < 360°
Solution:
- First, find the reference angle where sin(θ) = 1/2
- This occurs at θ = 30° (π/6 radians)
- Sine is positive in Quadrants I and II, so we have:
- θ = 30° (Quadrant I)
- θ = 180° - 30° = 150° (Quadrant II)
- Therefore, all solutions to sin(θ) = 1/2 for 0° ≤ θ < 360° are θ = 30° and θ = 150°
Unit Circle Quiz
Test your understanding of the unit circle with this interactive quiz!
Question 1: What are the coordinates of the point on the unit circle at 45 degrees?
Question 1 of 5
References and Tips
Memory Tips:
- ASTC Rule: "All Students Take Calculus" to remember which functions are positive in which quadrants.
- Special Angle Values: For angles 0°, 30°, 45°, 60°, 90°, the sine values follow the pattern: 0, 1/2, 1/√2, √3/2, 1.
- Complementary Angles: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ).
- 30-60-90 Triangle: The sides are in the ratio 1 : √3 : 2.
- 45-45-90 Triangle: The sides are in the ratio 1 : 1 : √2.
Common Mistakes to Avoid:
- Forgetting to account for quadrants when determining signs.
- Mixing up sine and cosine coordinates (remember: cosine is the x-coordinate, sine is the y-coordinate).
- Confusion between radians and degrees (always check which unit you're working with).
- Forgetting that tan(θ) = sin(θ)/cos(θ) and is undefined when cos(θ) = 0.
- Not using exact values for special angles (using decimals instead of exact expressions like √3/2).