Mathematics.

trigonometric functions

Unit Circle

Trigonometry35 minDifficulty3 out of 10

You should know: functions, pythagorean theorem

Overview

The unit circle is a circle of radius 1 centered at the origin. It provides the modern definition of sine and cosine for ANY angle — not just angles inside a right triangle — by relating an angle to the coordinates of the point it sweeps out on the circle. This single idea extends trigonometry from acute angles in triangles to all real numbers, including negative angles and angles greater than 360°.

Intuition

Picture a point starting at (1, 0) and walking counterclockwise around a circle of radius 1. As it walks, it sweeps out an angle θ from the positive x-axis. At every moment, its x-coordinate IS cos(θ) and its y-coordinate IS sin(θ) — by definition, not by triangle geometry. This immediately explains why sine and cosine are periodic (walking all the way around brings you back to the start) and why they're bounded between -1 and 1 (you can never leave the circle).

Interactive Graph

Drag the angle around the unit circle and watch sin/cos update

Loading visualization…

Formal Definition

Definition

For an angle θ measured counterclockwise from the positive x-axis, the point where the terminal ray meets the unit circle has coordinates:

(cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta)

The defining coordinates of the point on the unit circle at angle θ

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

The equation of the unit circle itself, giving the Pythagorean identity

Notation

NotationMeaning
θ\thetaThe angle, usually measured in radians
rad\text{rad}Radians — the angle subtending an arc equal in length to the radius; 2π radians = 360°

Properties

Pythagorean identity

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

Periodicity

sin⁡(θ+2π)=sin⁡θ,cos⁡(θ+2π)=cos⁡θ\sin(\theta + 2\pi) = \sin\theta,\quad \cos(\theta+2\pi)=\cos\theta

Even/odd symmetry

cos⁡(−θ)=cos⁡θ (even),sin⁡(−θ)=−sin⁡θ (odd)\cos(-\theta)=\cos\theta \text{ (even)},\quad \sin(-\theta)=-\sin\theta \text{ (odd)}

Key values

sin⁡(0)=0, sin⁡(π/2)=1, cos⁡(0)=1, cos⁡(π/2)=0\sin(0)=0,\ \sin(\pi/2)=1,\ \cos(0)=1,\ \cos(\pi/2)=0

Example: θ = π/6, π/4, π/3 give the classic 30-45-60 degree values

Applications

Simple harmonic motion (springs, pendulums) and wave phenomena are described directly using sine and cosine derived from the unit circle.

Worked Examples

  1. π/2 radians = 90°, which points straight up to (0, 1).

    (cos⁡(π/2),sin⁡(π/2))=(0,1)(\cos(\pi/2), \sin(\pi/2)) = (0, 1)

Answer: sin(π/2) = 1, cos(π/2) = 0

Practice Problems

Difficulty 3/10

What is cos(π) on the unit circle?

Difficulty 4/10

A wheelchair ramp rises 1 m over a horizontal run of 12 m. Using trigonometry, what angle does it make with the ground, and does it meet a common 1:12 (≈4.76°) accessibility guideline?

Difficulty 5/10

In CAD and robotics, a point is rotated about the origin using (cosθ, sinθ) from the unit circle. Rotate the point (1, 0) by 90° counter-clockwise. What are the new coordinates?

Difficulty 4/10

Find the exact value of sin(210°) using the unit circle and reference angles.

Common Mistakes

Common Mistake

Mixing up degrees and radians when evaluating trig functions.

sin(90) in radian mode ≠ sin(90°). Always confirm which angular unit your calculator or formula expects — most calculus uses radians exclusively.

Quiz

On the unit circle, what does the y-coordinate of the point at angle θ represent?

Flashcards

1 / 2

Historical Background

Trigonometry began with chord tables for astronomy: Hipparchus (2nd century BCE) is often credited as the father of trigonometry for his table of chords. The Indian mathematician Aryabhata (5th century CE) introduced the half-chord, essentially our modern sine, and Islamic Golden Age mathematicians including Al-Battani and Abu al-Wafa refined the six trigonometric ratios. The unit-circle framing that unifies these ratios as coordinates is a comparatively modern pedagogical device, standardized in the 20th century.

  1. c. 140 BCE

    Hipparchus compiles the first trigonometric table (of chords)

    Hipparchus

  2. c. 500 CE

    Aryabhata defines the half-chord (jya), precursor to sine

    Aryabhata

  3. 9th–10th century

    Islamic Golden Age scholars define all six trig ratios

    Al-Battani, Abu al-Wafa

Summary

  • The unit circle has radius 1, centered at the origin.
  • A point at angle θ (counterclockwise from positive x-axis) has coordinates (cos θ, sin θ).
  • This extends sine/cosine beyond right-triangle geometry to all real angles.
  • Pythagorean identity sin²θ+cos²θ=1 falls straight out of the circle equation x²+y²=1.
  • Periodicity (period 2π) and boundedness (-1 to 1) are immediate consequences of the circular definition.

References

  1. WebsiteWikipedia — Unit circle