Welcome to Trigonometry, Unit Circle Training
Introduction to the Unit Circle
The unit circle is a powerful tool in trigonometry that helps us understand the relationship between angles and the trigonometric functions (sine, cosine, and tangent). It's called the "unit" circle because it has a radius of exactly 1.
The unit circle is a circle with:
- Center at the origin (0, 0)
- Radius of 1 unit
- Equation: x² + y² = 1
Why Use the Unit Circle?
The unit circle allows us to:
- Visualize angles in standard position
- Find exact values of trigonometric functions
- Understand the periodic nature of trig functions
- Solve right triangle problems efficiently
Angles and Coordinates
Every point on the unit circle corresponds to an angle measured from the positive x-axis (moving counterclockwise). The coordinates of each point (x, y) give us the cosine and sine of that angle:
cos(θ) = x
sin(θ) = y
tan(θ) = y/x
Understanding Angles: Degrees and Radians
Angles can be measured in two ways: degrees and radians. While degrees are more familiar in everyday use, radians are the standard in mathematics and trigonometry.
Degrees
A complete circle is divided into 360 degrees (360°). This system dates back to ancient Babylonian astronomy.
Radians
Radians measure angles based on the radius of the circle. One radian is the angle created when the arc length equals the radius.
• Full circle = 360° = 2π radians
• Half circle = 180° = π radians
• Quarter circle = 90° = π/2 radians
• One radian ≈ 57.3°
Degrees to Radians: radians = degrees × (π/180)
Radians to Degrees: degrees = radians × (180/π)
Example 1:
Convert 60° to radians:60° × (π/180) = π/3 radians
Example 2:
Convert π/4 radians to degrees:(π/4) × (180/π) = 45°
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 |
📝 Quick Check: Angle Conversions
Question 1: Convert 90° to radians
Question 2: Convert π/6 radians to degrees
Key Angles on the Unit Circle
Certain angles appear frequently in trigonometry. These "special angles" have exact values that are worth memorizing.
The Special Angles: 30°, 45°, and 60°
0° (0 radians): (1, 0) → cos = 1, sin = 0
30° (π/6): (√3/2, 1/2) → cos = √3/2, sin = 1/2
45° (π/4): (√2/2, √2/2) → cos = √2/2, sin = √2/2
60° (π/3): (1/2, √3/2) → cos = 1/2, sin = √3/2
90° (π/2): (0, 1) → cos = 0, sin = 1
Pattern Recognition
Notice the pattern in the sine values for 0°, 30°, 45°, 60°, and 90°:
sin(30°) = √1/2 = 1/2
sin(45°) = √2/2
sin(60°) = √3/2
sin(90°) = √4/2 = 1
For cosine, the pattern is reversed!
The Four Quadrants
The unit circle is divided into four quadrants. The signs of sine and cosine change depending on the quadrant:
| Quadrant | Angle Range | cos (x) | sin (y) |
|---|---|---|---|
| I | 0° to 90° | Positive | Positive |
| II | 90° to 180° | Negative | Positive |
| III | 180° to 270° | Negative | Negative |
| IV | 270° to 360° | Positive | Negative |
Example:
Find cos(150°) and sin(150°):150° is in Quadrant II (reference angle: 180° - 150° = 30°)
cos(150°) = -cos(30°) = -√3/2 (negative in Q II)
sin(150°) = sin(30°) = 1/2 (positive in Q II)
📝 Quick Check: Unit Circle Values
Question 1: What is sin(45°)?
Question 2: What is cos(60°)?
Question 3: In which quadrant are both sine and cosine negative?
Solving Right Triangles Using the Unit Circle
The unit circle concepts directly apply to solving real-world right triangle problems. Let's see how!
Right Triangle Basics
sin(θ) = Opposite / Hypotenuse = y / r
cos(θ) = Adjacent / Hypotenuse = x / r
tan(θ) = Opposite / Adjacent = y / x
Connection to the Unit Circle
When the hypotenuse (r) equals 1 (as in the unit circle), the formulas simplify beautifully:
cos(θ) = x / 1 = x (horizontal position on unit circle)
tan(θ) = y / x (slope of the line)
Solving for Unknown Sides
If you know one angle (besides the right angle) and one side, you can find all other sides!
Example 1: Finding the Opposite Side
Given: θ = 30°, Hypotenuse = 10Find: Opposite side
Solution:
sin(30°) = Opposite / 10
0.5 = Opposite / 10
Opposite = 10 × 0.5 = 5 units
Example 2: Finding the Hypotenuse
Given: θ = 45°, Adjacent = 8Find: Hypotenuse
Solution:
cos(45°) = 8 / Hypotenuse
√2/2 ≈ 0.707 = 8 / Hypotenuse
Hypotenuse = 8 / 0.707 ≈ 11.31 units
Example 3: Finding an Angle
Given: Opposite = 6, Adjacent = 8Find: Angle θ
Solution:
tan(θ) = 6 / 8 = 0.75
θ = arctan(0.75) ≈ 36.87°
The Pythagorean Theorem
Don't forget this fundamental relationship in right triangles:
(Adjacent)² + (Opposite)² = (Hypotenuse)²
📝 Quick Check: Right Triangle Problems
Question 1: If θ = 60° and the hypotenuse = 12, what is the opposite side?
(Hint: sin(60°) = √3/2 ≈ 0.866)
Question 2: If the adjacent side = 5 and hypotenuse = 10, what is cos(θ)?
Question 3: In a right triangle, if opposite = 3 and adjacent = 4, what is the hypotenuse?
📋 Final Assessment
Test your knowledge of unit circle trigonometry with this comprehensive assessment. Good luck!