Unit Circle - Definition, Examples, Quiz, FAQ, Trivia

What is the Unit Circle?

The unit circle is a special circle with a radius of exactly 1 unit. It's centered at the origin point (0,0) on a coordinate plane. This simple circle helps us understand angles and trigonometric functions like sine, cosine, and tangent.

Why is it called "unit"? Because the radius is one unit long. This makes calculations easier because we can use simple fractions instead of complicated numbers.

The unit circle connects geometry with trigonometry. Every point on the circle has coordinates (x, y) that represent cosine and sine values for the angle formed with the positive x-axis.

How the Unit Circle Works

The unit circle uses two ways to measure angles: degrees and radians. Degrees are what you see on a protractor (0° to 360°). Radians are another way to measure angles based on the circle itself.

Key angle measurements:

• 0° (0 radians) - Point at (1, 0)
• 90° (π/2 radians) - Point at (0, 1)
• 180° (π radians) - Point at (-1, 0)
• 270° (3π/2 radians) - Point at (0, -1)

Trigonometric Functions

The unit circle helps us define the three main trigonometric functions:

Sine (sin): The y-coordinate of the point on the unit circle
Cosine (cos): The x-coordinate of the point on the unit circle
Tangent (tan): The ratio of sine to cosine (y/x)

These functions help us understand relationships between angles and sides of triangles. The unit circle makes it easy to remember important values:

Unit Circle Chart

The unit circle chart shows all the important angles and their corresponding sine, cosine, and tangent values. Here's a simplified version:

Real-World Examples

Let's see how the unit circle helps solve real problems:

Example 1: Find the sine and cosine of 45°
Solution: From the unit circle, at 45° (π/4 radians), both coordinates are √2/2. So sin(45°) = √2/2 and cos(45°) = √2/2.

Example 2: What is the tangent of 60°?
Solution: At 60° (π/3 radians), sin(60°) = √3/2 and cos(60°) = 1/2. Tan = sin/cos = (√3/2) ÷ (1/2) = √3.

Example 3: Verify the Pythagorean identity at 30°
Solution: sin(30°) = 1/2, cos(30°) = √3/2. sin² + cos² = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1. It works!

Example 4: Convert 90° to radians
Solution: Radians = Degrees × (π/180) = 90 × π/180 = π/2.

Unit Circle Practice Quiz

Test your understanding with this 5-question quiz. Choose the correct answer for each question.

Frequently Asked Questions

Here are answers to common questions about the unit circle:

Math Trivia

Discover interesting facts about trigonometry and circles: