The Unit Circle
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What Is the Unit Circle?
Definition: Unit Circle
The unit circle is a circle with radius 1 centered at the origin. Its equation is:
x² + y² = 1
The unit circle is the foundation for understanding trigonometric functions beyond right triangles. It allows us to define sine, cosine, and tangent for any angle, including angles greater than 90° and negative angles.
Angles on the Unit Circle
Radians vs. Degrees
Angles can be measured in degrees or radians. On the unit circle, radians are often preferred.
Radian Measure
One radian is the angle formed when the arc length equals the radius.
360° = 2π radians
180° = π radians
Conversion:
- Degrees to radians: multiply by π/180
- Radians to degrees: multiply by 180/Ï€
Common Angles
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Coordinates on the Unit Circle
For any angle θ measured from the positive x-axis:
Unit Circle Definition of Sine and Cosine
If the terminal side of angle θ intersects the unit circle at point (x, y), then:
cos θ = x (the x-coordinate)
sin θ = y (the y-coordinate)
tan θ = y/x = sin θ / cos θ (when x ≠0)
The Special Angles
Memorize these coordinates - they're the foundation of the unit circle:
Quadrant I (0° to 90°)
| Angle | Coordinates (cos θ, sin θ) |
|---|---|
| 0° or 0 | (1, 0) |
| 30° or π/6 | (√3/2, 1/2) |
| 45° or π/4 | (√2/2, √2/2) |
| 60° or π/3 | (1/2, √3/2) |
| 90° or π/2 | (0, 1) |
The Four Quadrants
Signs of trig functions depend on the quadrant:
ASTC Rule (All Students Take Calculus)
- Quadrant I: All positive
- Quadrant II: Sine positive (cos and tan negative)
- Quadrant III: Tangent positive (sin and cos negative)
- Quadrant IV: Cosine positive (sin and tan negative)
Reference Angles
Reference Angle
The reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It's always between 0° and 90° (or 0 and π/2).
To find trig values for any angle:
- Find the reference angle
- Determine the trig values for the reference angle
- Apply the correct sign based on the quadrant
Complete Unit Circle
Here are the coordinates for all major angles:
| Quadrant | Angle | (cos, sin) |
|---|---|---|
| I | 0° | (1, 0) |
| 30° | (√3/2, 1/2) | |
| 45° | (√2/2, √2/2) | |
| 60° | (1/2, √3/2) | |
| II | 90° | (0, 1) |
| 120° | (-1/2, √3/2) | |
| 135° | (-√2/2, √2/2) | |
| 150° | (-√3/2, 1/2) | |
| III | 180° | (-1, 0) |
| 210° | (-√3/2, -1/2) | |
| 225° | (-√2/2, -√2/2) | |
| 240° | (-1/2, -√3/2) | |
| IV | 270° | (0, -1) |
| 300° | (1/2, -√3/2) | |
| 315° | (√2/2, -√2/2) | |
| 330° | (√3/2, -1/2) |
Examples
Example 1: Finding Coordinates
Problem: Find the exact coordinates of the point on the unit circle at angle 5Ï€/6.
Solution:
Step 1: Convert to degrees: 5π/6 × (180/π) = 150°
Step 2: Find quadrant: 150° is in Quadrant II
Step 3: Find reference angle: 180° - 150° = 30°
Step 4: Coordinates for 30°: (√3/2, 1/2)
Step 5: In QII, x is negative, y is positive
Point at 5π/6: (-√3/2, 1/2)
Example 2: Evaluating Trig Functions
Problem: Find sin(240°) and cos(240°).
Solution:
Step 1: 240° is in Quadrant III
Step 2: Reference angle: 240° - 180° = 60°
Step 3: sin(60°) = √3/2 and cos(60°) = 1/2
Step 4: In QIII, both sine and cosine are negative
sin(240°) = -√3/2
cos(240°) = -1/2
Example 3: Converting Units
Problem: Convert 225° to radians. Convert 7π/4 to degrees.
Solution:
225° to radians: 225 × (π/180) = 225π/180 = 5π/4
7π/4 to degrees: (7π/4) × (180/π) = 7(180)/4 = 315°
Example 4: Finding Tangent
Problem: Find tan(5Ï€/3).
Solution:
Step 1: 5π/3 = 300° (Quadrant IV)
Step 2: Reference angle: 360° - 300° = 60°
Step 3: At 300°: cos = 1/2, sin = -√3/2
Step 4: tan = sin/cos = (-√3/2)/(1/2) = -√3
Example 5: Negative Angles
Problem: Find cos(-Ï€/3).
Solution:
Step 1: -π/3 = -60° (measured clockwise)
Step 2: Same as 360° - 60° = 300° (Quadrant IV)
Step 3: Reference angle is 60°
Step 4: cos(60°) = 1/2, and cosine is positive in QIV
cos(-Ï€/3) = 1/2
Practice
1. Convert 270° to radians.
2. Convert 5Ï€/4 to degrees.
3. Find the coordinates of the point on the unit circle at angle π/3.
4. Find sin(150°) and cos(150°).
5. Find tan(315°).
6. In which quadrant is the angle 7Ï€/6?
7. What is the reference angle for 200°?
8. Find sin(-Ï€/6).
9. If cos θ = -1/2 and θ is in Quadrant II, find θ in radians.
10. Find all angles θ in [0, 2π) where sin θ = √2/2.
Click to reveal answers
- 3Ï€/2
- 225°
- (1/2, √3/2)
- sin(150°) = 1/2; cos(150°) = -√3/2
- tan(315°) = -1
- Quadrant III (7π/6 = 210°)
- 200° - 180° = 20°
- sin(-Ï€/6) = -1/2
- θ = 2π/3 (120°)
- θ = π/4 and θ = 3π/4 (45° and 135°)
Check Your Understanding
1. Why do we use the unit circle (radius 1) rather than any other size circle?
Show answer
With radius 1, the x-coordinate equals cos θ and the y-coordinate equals sin θ directly, without any scaling needed. This is because in a right triangle with hypotenuse 1, adjacent/hypotenuse = adjacent = cos θ and opposite/hypotenuse = opposite = sin θ.
2. Why is sine positive in Quadrants I and II, but negative in III and IV?
Show answer
Sine equals the y-coordinate on the unit circle. In Quadrants I and II, points are above the x-axis, so y > 0 (sine positive). In Quadrants III and IV, points are below the x-axis, so y < 0 (sine negative).
3. How can you quickly determine the trig values for 135° if you know the values for 45°?
Show answer
135° and 45° have the same reference angle (45°), so they have the same absolute values. 135° is in Quadrant II where sine is positive and cosine is negative. So sin(135°) = sin(45°) = √2/2, but cos(135°) = -cos(45°) = -√2/2.
4. What is the relationship between sin θ and sin(-θ)? Between cos θ and cos(-θ)?
Show answer
sin(-θ) = -sin θ (sine is an odd function - changes sign)
cos(-θ) = cos θ (cosine is an even function - stays the same)
This is because rotating clockwise gives the same x-coordinate but opposite y-coordinate.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review