Precalculus and Trigonometry · 35 minutes
Suggested grades 11–12 · Unit 2: Exponential inverses and the unit circle · Lesson 2 of 2
The Unit Circle Beyond One Turn
Interpret sine and cosine as coordinates for positive and negative radian angles.
Mr. S Teaches · Original starter lesson · HS-PRE-04 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to the unit circle | Trigonometry | Khan Academy
By Khan Academy
Extend right-triangle ratios to coordinate-based trigonometric values.
Open on YouTube: Introduction to the unit circle | Trigonometry | Khan Academy (opens a new tab) ↗Unit Circle Definition of Trig Functions
By Khan Academy
Compare the coordinate definitions of sine and cosine before considering additional complete turns.
Open on YouTube: Unit Circle Definition of Trig Functions (opens a new tab) ↗Radian and degree | Unit circle definition of trig functions | Trigonometry | Khan Academy
By Khan Academy
Review degree-radian conversion when representing angles beyond one revolution.
Open on YouTube: Radian and degree | Unit circle definition of trig functions | Trigonometry | Khan Academy (opens a new tab) ↗Why do coterminal angles have the same sine and cosine coordinates?
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Learning objective
Locate coterminal angles and determine coordinate-based sine and cosine values.
Before you begin
- Coordinate plane
- Radian measure
- Circle radius
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- unit circle
- A radius-one circle centered at the origin.
- coterminal angles
- Angles ending at the same ray after whole turns.
- radian measure
- Arc length per unit radius, with 2π for one full turn.
Understand the concept
On the unit circle, angleθ corresponds to coordinates(cosθ,sinθ). Counterclockwise rotation is positive; clockwise is negative. Adding 2π changes the number of turns but not the terminal point, allowing trigonometric functions to accept angles beyond an acute triangle. Coordinate signs follow the quadrant.
Teach it step by step
- Draw the unit circle and label quarter-turn points at 0,π/2,π,3π/2 and 2π.
- Locate 5π/2 by removing a full turn and identify its coordinates.
- Locate−π/2 by moving clockwise and compare it with 3π/2.
- For a supplied point(−3/5,4/5) in quadrantII, state sine and cosine and verify it lies on the circle.
Make the thinking visible
Worked example or model response
5π/2 is coterminal withπ/2, so cosine 0 and sine 1. −π/2 corresponds to(0,−1). For(−3/5,4/5), cosine=−3/5 and sine=4/5;9/25+16/25=1.
Guided practice
Find coordinates, sine and cosine for−π,7π/2 and 4π.
Independent practice
1. Give a coterminal angle in[0,2π) for−5π/2. 2. A unit-circle point is(5/13,−12/13). Identify its quadrant and sine.
Quick assessment
Why does a coordinate of−1 for sine not imply a negative circle radius?
Answer key & assessment criteria
Answers and teaching guidance
- Guided:−π→(−1,0);7π/2→(0,−1);4π→(1,0). Sine is each y-coordinate and cosine each x-coordinate.
- Independent 1:3π/2. 2: QuadrantIV; sine−12/13.
- Quick check: Sine is a signed vertical coordinate; radius is the nonnegative distance from the center.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use a paper arrow for rotation direction and count quarter turns before general angles.
Extension challenge
Explain why every integer multiple of 2π returns to the same point, including negative multiples.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.TF.A.2
Common Core · High school · Functions
Regular high school mathematics standard
Use unit-circle coordinates and radian rotations to extend trigonometric functions.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.