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5.1 The Unit Circle

Connect angle measure to coordinates.

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Ideas in this lesson

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01Unit-circle equation and membershipWorked steps · five questions 02Recover signed missing coordinateWorked steps · five questions 03Signed real arc and terminal pointWorked steps · five questions 04Reference numbersWorked steps · five questions 05Read approximate terminal coordinatesWorked steps · five questions 06Transform a known terminal pointWorked steps · five questions 07Derive special terminal pointsWorked steps · five questions

Lesson brief and rules

5.1 The Unit Circle

On the unit circle, $\cos \theta$ is the $x$ -coordinate and $\sin \theta$ is the $y$ -coordinate. One full revolution is $2\pi$ radians.

Remember
Use radians unless an angle is marked in degrees.

Worked example

1

01 / A circle consists of points at distance

1

from the origin. State its center, radius and equation. Test

\left(\dfrac{4}{5},\dfrac{3}{5}\right)

and

\left(\dfrac{4}{5},\dfrac{4}{5}\right)

by exact squared sums, not decimal estimates. Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

02 /

x^{2}+y^{2}=1

follows from the distance formula with center

\left(0,0\right)

and radius

1

03 /

\dfrac{16}{25}+\dfrac{9}{25}=1

verifies the first point.

04 /

\dfrac{16}{25}+\dfrac{16}{25}=\dfrac{32}{25}

, which differs from

1

, so the second point is not on the circle.

Worked example 2 · Unit-circle equation and membership

A circle consists of points at distance $1$ from the origin. State its center, radius and equation. Test $\left(\dfrac{2}{5},- \dfrac{\sqrt{21}}{5}\right)$ and $\left(\dfrac{2}{5},\dfrac{1}{6} - \dfrac{\sqrt{21}}{5}\right)$ by exact squared sums, not decimal estimates. Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

$x^{2}+y^{2}=1$ follows from the distance formula with center $\left(0,0\right)$ and radius $1$ .
$\dfrac{4}{25}+\dfrac{21}{25}=1$ verifies the first point.
$\dfrac{4}{25}+\dfrac{781}{900} - \dfrac{\sqrt{21}}{15}=\dfrac{37}{36} - \dfrac{\sqrt{21}}{15}$ , which differs from $1$ , so the second point is not on the circle.
Written-work review:

Use the stated definitions and the complete allowed domain.
Show all intermediate values and justify signs, restrictions and conclusions.
Give every requested result in the stated order and units.

Read the graphs

Signed real arc and terminal point

Starting with A on the positive horizontal axis, the marked points are separated by

\dfrac{\pi }{4}

. Give the input in

[0,2\pi )

and both exact coordinates of every point. Explain why

2\pi

returns to A but is not a second listed input.

Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.

Each counterclockwise step adds $\dfrac{\pi }{4}$ ; $P(t+2\pi )=P(t)$ .

Point	Input	Coordinates
$A$	$0$	$\left(1,0\right)$
$B$	$\dfrac{\pi }{4}$	$\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$
$C$	$\dfrac{\pi }{2}$	$\left(0,1\right)$
$D$	$\dfrac{3 \pi }{4}$	$\left(- \dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$
$E$	$\pi$	$\left(-1,0\right)$
$F$	$\dfrac{5 \pi }{4}$	$\left(- \dfrac{\sqrt{2}}{2},- \dfrac{\sqrt{2}}{2}\right)$
$G$	$\dfrac{3 \pi }{2}$	$\left(0,-1\right)$
$H$	$\dfrac{7 \pi }{4}$	$\left(\dfrac{\sqrt{2}}{2},- \dfrac{\sqrt{2}}{2}\right)$

Positive input means counterclockwise arc length on radius 1; negative input means clockwise. The circumference is

2\pi

.
Written-work review:

Use the stated definitions and the complete allowed domain.
Show all intermediate values and justify signs, restrictions and conclusions.
Give every requested result in the stated order and units.

Answer: In counterclockwise order, $\left(1,0\right)$ ; $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$ ; $\left(0,1\right)$ ; $\left(- \dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$ ; $\left(-1,0\right)$ ; $\left(- \dfrac{\sqrt{2}}{2},- \dfrac{\sqrt{2}}{2}\right)$ ; $\left(0,-1\right)$ ; $\left(\dfrac{\sqrt{2}}{2},- \dfrac{\sqrt{2}}{2}\right)$ .

Read approximate terminal coordinates

The plotted terminal points

A

and

B

correspond to

t=\dfrac{61}{100}

and

t=- \dfrac{131}{100}

. Use the actual diagram to estimate both coordinates to one decimal; do not begin with a calculator. Explain the signs and direction.

Read horizontal and vertical projections against the grid, in that order. $A$ is $\left(0.8,0.6\right)$ , $B$ is $\left(0.3,-1.0\right)$ .
The positive input is counterclockwise; the negative input is clockwise. Coordinate signs agree with the visible quadrants.
Calculator audit, not the estimation method: $\left(0.81965,0.57287\right)$ and $\left(0.25785,-0.96618\right)$ .
Written-work review:

Use the stated definitions and the complete allowed domain.
Show all intermediate values and justify signs, restrictions and conclusions.
Give every requested result in the stated order and units.

Answer: $0.8$ , $0.6$ , $0.3$ , $-1.0$

Derive special terminal points

Derive the diagonal point using symmetry and the circle equation, without quoting a memorized special-angle table. Target input:

\dfrac{\pi }{4}

. Then explain what changes if the construction is performed on a circle of radius

2

. The diagram supplies geometry, not the coordinate proof.

For the first-quadrant diagonal, symmetry gives $y=x$ . Substitute into $x^{2}+y^{2}=1$ to obtain $2x^{2}=1$ , hence $x=y=\dfrac{\sqrt{2}}{2}$ because both are positive.
The requested diagonal has signs fixed by its quadrant, giving $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$ .
Scaling the diagram by $2$ multiplies both coordinates and all chord lengths by $2$ but leaves the angle unchanged; normalized back to radius $1$ , the same answer results.
Written-work review:

Justify equal arcs give equal chords rather than assume the coordinates.
Write and solve the distance and circle equations, rejecting the extraneous quadrant root.
Explain the diagonal and axis reflections and the radius-scaling conclusion.

Answer: $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)$

15 questions with a worked answer for each.Start practice ↗

Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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MathLab with Mr. Ahmed

5.1 The Unit Circle

Ideas in this lesson

5.1 The Unit Circle

Read the graphs

Signed real arc and terminal point

Read approximate terminal coordinates

Derive special terminal points