Lesson 1: Defining Sine and Cosine on the Unit Circle
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define the unit circle and explain its importance in trigonometry
- Define sine and cosine as coordinates of a point on the unit circle
- Determine the sign of sine and cosine in each quadrant
- Evaluate sine and cosine at quadrantal angles (0, π/2, π, 3π/2)
- State the Pythagorean identity sin²θ + cos²θ = 1
What Is the Unit Circle?
The unit circle is a circle with radius 1, centered at the origin of the coordinate plane. It provides the foundation for defining trigonometric functions for any angle, not just acute angles in right triangles.
Unit Circle — The circle x² + y² = 1, centered at (0, 0) with radius 1.
To connect angles to the unit circle, start at the point (1, 0) and move counterclockwise around the circle by angle θ. The point where you land has coordinates (x, y).
Defining Sine and Cosine
The coordinates of the point on the unit circle corresponding to angle θ define sine and cosine.
Unit Circle Definitions:
If angle θ (in standard position) intersects the unit circle at point (x, y), then:
cos θ = x (the horizontal coordinate)
sin θ = y (the vertical coordinate)
This means: cosine tells you how far right or left the point is, and sine tells you how far up or down.
Example 1: Understanding the Definition
The angle θ = 0 corresponds to the point (1, 0) on the unit circle.
Therefore: cos(0) = 1 and sin(0) = 0
Example 2: Angle π/2 (90°)
Moving counterclockwise by 90° from (1, 0) brings us to the top of the circle: (0, 1).
Therefore: cos(π/2) = 0 and sin(π/2) = 1
Evaluating at Quadrantal Angles
The quadrantal angles are those whose terminal side lies on an axis: 0, π/2, π, 3π/2, and 2π.
| Angle θ | Point (x, y) | cos θ | sin θ |
|---|---|---|---|
| 0 | (1, 0) | 1 | 0 |
| π/2 | (0, 1) | 0 | 1 |
| π | (−1, 0) | −1 | 0 |
| 3π/2 | (0, −1) | 0 | −1 |
| 2π | (1, 0) | 1 | 0 |
Signs of Sine and Cosine by Quadrant
Since cos θ = x and sin θ = y, the signs of sine and cosine depend on the quadrant:
Quadrant I: sin > 0, cos > 0 (both coordinates positive)
Quadrant II: sin > 0, cos < 0 (x negative, y positive)
Quadrant III: sin < 0, cos < 0 (both negative)
Quadrant IV: sin < 0, cos > 0 (x positive, y negative)
A common mnemonic: "All Students Take Calculus" — All trig functions positive in QI, Sine in QII, Tangent in QIII, Cosine in QIV.
Example 3: Determining Sign
Determine the sign of sin(200°) and cos(200°).
Solution: 200° is in Quadrant III (180° < 200° < 270°). In QIII, both sine and cosine are negative.
The Pythagorean Identity
Since (cos θ, sin θ) lies on the unit circle x² + y² = 1, we immediately get:
Pythagorean Identity: cos²θ + sin²θ = 1
This holds for every angle θ and is one of the most important identities in trigonometry.
Example 4: Using the Pythagorean Identity
If sin θ = 3/5 and θ is in Quadrant II, find cos θ.
Solution:
Step 1: cos²θ + sin²θ = 1, so cos²θ + (3/5)² = 1
Step 2: cos²θ + 9/25 = 1, so cos²θ = 16/25
Step 3: cos θ = ±4/5. Since θ is in QII, cosine is negative.
Answer: cos θ = −4/5
Domain and Range
Since (cos θ, sin θ) always lies on the unit circle:
- Domain of sine and cosine: all real numbers (any angle is valid)
- Range of sine and cosine: [−1, 1] (coordinates stay between −1 and 1)
Check Your Understanding
1. What are the coordinates of the point on the unit circle at θ = π?
2. In which quadrant is sine positive and cosine negative?
3. If cos θ = −5/13 and θ is in QIII, find sin θ.
Key Takeaways
- The unit circle has equation x² + y² = 1 with center (0,0) and radius 1.
- cos θ = x-coordinate and sin θ = y-coordinate of the point on the unit circle.
- Use "All Students Take Calculus" to remember sign patterns by quadrant.
- The Pythagorean identity cos²θ + sin²θ = 1 is fundamental.
- Both sine and cosine have domain (−∞, ∞) and range [−1, 1].
Ready for More?
Next Lesson
In Lesson 2 you will evaluate trig functions at special angles using the unit circle.
Start Lesson 2