Introduction to the Unit Circle

Learn what the unit circle is and how it connects angles to trigonometric values.

Advanced25 minLesson
Printable worksheet

Definition

The unit circle is a circle with a radius of exactly 11 unit, centered at the origin (0,0)(0, 0) of a coordinate plane.
Key Properties:
  • Center: (0,0)(0, 0)
  • Radius: r=1r = 1
  • Equation: x2+y2=1x^2 + y^2 = 1
For any point (x,y)(x, y) on the unit circle at angle θ\theta:
cos⁡(θ)=xandsin⁡(θ)=y\cos(\theta) = x \quad \text{and} \quad \sin(\theta) = y
This means the coordinates of any point on the unit circle are (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).

Try it now

What is the radius of the unit circle?

Worked Examples

What are the coordinates of the point on the unit circle at θ=0°\theta = 0°?

1

Locate the angle

Start at the positive x-axis and rotate 0°0° (no rotation) → Point is at the rightmost position

2

Find the x-coordinate (cosine)

The point is 1 unit to the right of the center → cos⁡(0°)=1\cos(0°) = 1

3

Find the y-coordinate (sine)

The point is 0 units above or below the center → sin⁡(0°)=0\sin(0°) = 0

Common Mistakes

Confusing which coordinate is sine and which is cosine

Why it's wrong: It's easy to mix up x and y when both relate to the angle.

Correct: Remember: Cosine = x-coordinate (horizontal), Sine = y-coordinate (vertical). Think 'x comes before y alphabetically, and cosine comes before sine.'

Forgetting that angles are measured counterclockwise

Why it's wrong: By convention, positive angles go counterclockwise from the positive x-axis.

Correct: Always start at the positive x-axis (3 o'clock position) and rotate counterclockwise for positive angles.

Using wrong signs in different quadrants

Why it's wrong: Students often forget that coordinates can be negative.

Correct: Use the ASTC rule: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4) - tells you which functions are positive.

Interactive Visual

Unit Circle

Degrees

0°

Radians

0

sin(θ)

0

cos(θ)

1

tan(θ)

0

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Right Triangle Trigonometry

θ =45°
5°45°85°
sin(θ)
√2/2
Opp / Hyp
cos(θ)
√2/2
Adj / Hyp
tan(θ)
1
Opp / Adj

Move the slider to change the angle and see how trigonometric ratios change.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy

What is the radius of the unit circle?

Why It Matters

The unit circle is the foundation of all trigonometry and appears throughout mathematics and science:
  • Physics: Describing circular motion, waves, and oscillations
  • Engineering: Analyzing alternating current (AC) circuits and signal processing
  • Computer Graphics: Rotating objects and creating smooth animations
  • Music: Understanding sound waves and harmonics
  • Navigation: GPS systems and satellite positioning
Once you master the unit circle, you can quickly find sine and cosine values without a calculator!

Real World Applications

Ferris Wheel Motion

A Ferris wheel rotates in a circle. Your height above ground follows a sine wave pattern as you ride!

Example:

On a Ferris wheel with radius 10 meters, your height varies between 0 and 20 meters as you complete each rotation.

1Try It Yourself

A Ferris wheel has a radius of 10 meters and its center is 12 meters above ground. You board at the bottom.

What is your height above ground when you've rotated 90 degrees?

Step 1: Write the mathematical expression

Height = center height + radius times sin(angle): 12+10×sin⁡(90°)12 + 10 \times \sin(90°)

Clock Hands

Clock hands rotate around the center, and their position can be described using the unit circle.

Example:

At 3:00, the minute hand points at 12 (top), which is the 90-degree position on the unit circle.

2Try It Yourself

A clock's minute hand has length 8 cm. At 15 minutes past the hour, the hand points to the right (3 on the clock).

What is the x-coordinate of the tip of the minute hand?

Step 1: Write the mathematical expression

x = radius times cos(angle). At 15 min, angle = 0°0°: 8×cos⁡(0°)8 \times \cos(0°)

Key Takeaways

  • 1The unit circle has radius 1 and is centered at the origin (0,0)(0, 0)
  • 2Any point on the unit circle has coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta)
  • 3Angles are measured counterclockwise from the positive x-axis
  • 4The equation of the unit circle is x2+y2=1x^2 + y^2 = 1
  • 5Key angles to memorize: 0°0°, 30°30°, 45°45°, 60°60°, 90°90° and their multiples

Frequently Asked Questions

It's called the unit circle because its radius is exactly 1 unit. This makes calculations simpler because multiplying by 1 doesn't change values.
It's called the unit circle because its radius is exactly 1 unit. This makes calculations simpler because multiplying by 1 doesn't change values.
Think alphabetically: 'c' comes before 's', and 'x' comes before 'y'. So cosine goes with x, and sine goes with y.
Both measure angles. Degrees divide a circle into 360 parts. Radians use the circle's radius: a full circle is 2π2\pi radians. To convert: 180°=π180° = \pi radians.

Glossary

Unit circle
A circle with radius 1 centered at the origin
Radian
An angle measure where 2π2\pi radians equals a full rotation (360°360°)
Reference angle
The acute angle formed between the terminal side of an angle and the x-axis
Quadrant
One of four regions of the coordinate plane, numbered I through IV counterclockwise from the upper right
Terminal side
The final position of a ray after rotation from the initial side (positive x-axis)

Formula Card

Circle Equation

x2+y2=1x^2 + y^2 = 1

Every point on the unit circle satisfies this equation

Coordinate Definition

(x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos\theta, \sin\theta)

Coordinates of a point at angle theta

Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Follows from the circle equation

Degree-Radian Conversion

θrad=θdeg×π180\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}

Convert degrees to radians

More in This Topic

Radians and Degrees

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