Use the unit circle to determine coordinates and angle relationships

Alberta Grade 12 Mathematics

Trigonometry

Alberta Mathematics 30-1 · TR2

A point on a circle can be described by its horizontal and vertical coordinates. On the unit circle, those coordinates have a direct connection to an angle. This lesson reviews how angles are measured, then shows how to use the circle to find coordinates and compare angles. Exact coordinates are given whenever possible; decimal values are identified as approximations.

What you will learn

Describe how an angle is placed on the unit circle.
Use the unit circle to find exact coordinates for common angles.
Use reference angles and symmetry to relate coordinates at different angles.
Identify when an angle has the same terminal arm as another angle.

1. Review: angles and coordinates

A coordinate is an ordered pair that gives a point's horizontal and vertical position. In the plane, a point is written as

(x,y)

. Positive

x

is to the right, and positive

y

is upward.

An angle in standard position begins on the positive

x

-axis. Its vertex is at the origin. A positive angle turns counterclockwise; a negative angle turns clockwise. The final ray is called the terminal arm.

Angles can be measured in degrees or radians. A full turn is

360^\circ

, which is also

2\pi

radians. Half a turn is

180^\circ

or

\pi

radians. These measures describe the same rotations in different units.

360^\circ=2\pi\text{ radians}

Positive angles turn counterclockwise; negative angles turn clockwise.
A full turn is $360^\circ=2\pi$ radians.

2. The unit circle and its coordinates

A unit circle is a circle with radius

1

and centre at the origin. For an angle in standard position, the terminal arm meets the circle at one point. That point's coordinates are determined by the angle.

If the angle is

\theta

and the point is

(x,y)

, then

x=\cos\theta

and

y=\sin\theta

. Therefore, the point can be written as

(\cos\theta,\sin\theta)

. This coordinate rule is the main tool in this lesson.

Some angles have especially useful exact coordinates. At

0^\circ

, the point is

(1,0)

. At

90^\circ

, it is

(0,1)

. At

180^\circ

, it is

(-1,0)

, and at

270^\circ

, it is

(0,-1)

. The same positions can be named using radians:

0

,

\frac{\pi}{2}

,

\pi

, and

\frac{3\pi}{2}

.

For angles between these axes, the common reference angles are

30^\circ

,

45^\circ

, and

60^\circ

. Their first-quadrant coordinates are, respectively,

(\frac{\sqrt{3}}{2},\frac{1}{2})

,

(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})

, and

(\frac{1}{2},\frac{\sqrt{3}}{2})

. The reference angle is the acute angle between the terminal arm and the horizontal axis. It helps identify the coordinate sizes in other quadrants.

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

The unit circle has centre $(0,0)$ and radius $1$ .
The point for angle $\theta$ is $(\cos\theta,\sin\theta)$ .
A reference angle is the acute angle between a terminal arm and the horizontal axis.

3. Signs, symmetry, and related angles

The coordinate signs depend on the quadrant. Quadrant I has positive

x

and positive

y

. Quadrant II has negative

x

and positive

y

. Quadrant III has both coordinates negative. Quadrant IV has positive

x

and negative

y

. The axes are not part of any quadrant.

Angles with the same reference angle have coordinate values with the same absolute sizes. Their signs change according to the quadrant. For example, an angle of

150^\circ

has reference angle

30^\circ

. Its horizontal coordinate is negative and its vertical coordinate is positive, so its coordinates are

(-\frac{\sqrt{3}}{2},\frac{1}{2})

.

Reflection across the vertical axis changes the sign of

x

but keeps

y

the same. Reflection across the horizontal axis keeps

x

the same but changes the sign of

y

. Reflection through the origin changes both signs. These symmetry patterns explain how coordinates for related angles are connected.

Angles that differ by a full turn end at the same point. Such angles are called coterminal angles. Add or subtract

360^\circ

in degree measure, or

2\pi

in radian measure, to find a coterminal angle. For instance,

-45^\circ

and

315^\circ

are coterminal, so they have the same coordinates.

Keep exact coordinates when the angle is a common special angle. A decimal approximation can be useful for checking, but it is not the exact answer. For angles that are not common special angles, a calculator may give approximate coordinates.

\theta+360° k (k∈\mathbb{Z})

Use the reference angle to find coordinate sizes, then use the quadrant to assign signs.
Coterminal angles differ by a whole turn and share a terminal point.
An exact value and a decimal approximation are not the same kind of answer.

4. A reliable method

First, place the angle in standard position and identify its quadrant. Next, find the reference angle. Use the unit-circle coordinates for that reference angle, then assign signs from the quadrant. If the angle is outside one full turn or is negative, first find a coterminal angle that is easier to read.

A quick check is to confirm that the point lies on the unit circle. Its distance from the origin must be

1

. In coordinate form, this means that the sum of the squares of the coordinates is

1

. This check can reveal a sign or value error.

Do not assume that a negative angle gives negative coordinates. The angle's direction determines its terminal arm; the quadrant determines the coordinate signs.

x^2+y^2=1

Locate the angle, find its reference angle, and assign signs.
Check that the coordinates satisfy the unit-circle condition.

Reference coordinates and quadrant signs

Reference angle	First-quadrant coordinates	Quadrant II signs	Quadrant III signs	Quadrant IV signs
$30^\circ$	$(\frac{\sqrt{3}}{2},\frac{1}{2})$	$(-,+)$	$(-,-)$	$( +,-)$
$45^\circ$	$(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$	$(-,+)$	$(-,-)$	$( +,-)$
$60^\circ$	$(\frac{1}{2},\frac{\sqrt{3}}{2})$	$(-,+)$	$(-,-)$	$( +,-)$

Worked example

Find coordinates in Quadrant II

Determine the exact coordinates on the unit circle for an angle of

120^\circ

.

Angle at 120 degrees

The point for $120^\circ$ lies in Quadrant II.

Locate the angle
The angle is between $90^\circ$ and $180^\circ$ , so its terminal arm is in Quadrant II. The $x$ -coordinate will be negative and the $y$ -coordinate positive.
Find the reference angle
In Quadrant II, subtract the angle from $180^\circ$ . This gives the acute angle from the terminal arm to the negative horizontal axis.
$180^\circ-120^\circ=60^\circ$
Use the reference coordinates
The first-quadrant point for $60^\circ$ is $(\frac{1}{2},\frac{\sqrt{3}}{2})$ . Apply the Quadrant II signs: negative $x$ , positive $y$ .
$\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

Answer: The exact coordinates are

\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)

.

Check: The point is in Quadrant II, as required. Also,

(-\frac{1}{2})^2+(\frac{\sqrt{3}}{2})^2=\frac{1}{4}+\frac{3}{4}=1

.

Worked example

Use a coterminal angle

Determine the exact coordinates for an angle of

-225^\circ

.

Coterminal angle at 135 degrees

The angle $-225^\circ$ has the same terminal point as $135^\circ$.

Find an easier coterminal angle
Add one full turn. The resulting positive angle has the same terminal arm as the given negative angle.
$-225^\circ+360^\circ=135^\circ$
Find the reference angle
The coterminal angle is in Quadrant II. Its reference angle is the difference between it and $180^\circ$ .
$180^\circ-135^\circ=45^\circ$
Assign the coordinate signs
The $45^\circ$ point has equal coordinate sizes, each $\frac{\sqrt{2}}{2}$ . In Quadrant II, $x$ is negative and $y$ is positive.
$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

Answer: The exact coordinates for

-225^\circ

are

\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)

.

Check: The coterminal angle

135^\circ

is in Quadrant II. The coordinate squares sum to

\frac{1}{2}+\frac{1}{2}=1

.

Worked example

Compare related angles

An angle has coordinates

\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)

. Give one angle in each of Quadrants II, III, and IV that has the same reference angle, and state the coordinates for each.

Angles with a 30-degree reference angle

The three points have the same reference angle and coordinate sizes.

Identify the reference angle
The given coordinates match the unit-circle point for $30^\circ$ . Therefore, the reference angle is $30^\circ$ .
$\left(\cos 30^\circ,\sin 30^\circ\right)=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$
Find the Quadrant II angle
In Quadrant II, subtract the reference angle from $180^\circ$ . Keep the coordinate sizes and make only the horizontal coordinate negative.
$180^\circ-30^\circ=150^\circ,\quad \left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$
Find the Quadrant III angle
In Quadrant III, add the reference angle to $180^\circ$ . Both coordinates are negative.
$180^\circ+30^\circ=210^\circ,\quad \left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$
Find the Quadrant IV angle
In Quadrant IV, subtract the reference angle from $360^\circ$ . The horizontal coordinate is positive and the vertical coordinate is negative.
$360^\circ-30^\circ=330^\circ,\quad \left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$

Answer: The angles and coordinates are

150^\circ: (-\frac{\sqrt{3}}{2},\frac{1}{2})

,

210^\circ: (-\frac{\sqrt{3}}{2},-\frac{1}{2})

, and

330^\circ: (\frac{\sqrt{3}}{2},-\frac{1}{2})

.

Check: Each point has the same coordinate sizes as the given point. The signs match the stated quadrant.

Common mistakes and how to avoid them

Giving the same positive coordinates in every quadrant.

Correction: Use the reference angle for the coordinate sizes, then set each sign from the quadrant.

Treating a negative angle as if both coordinates must be negative.

Correction: Find a positive coterminal angle or locate the terminal arm. Coordinate signs depend on the terminal arm's quadrant.

Calling a decimal approximation an exact coordinate.

Correction: Keep radicals and fractions for common special angles. Label calculator decimals as approximations.

Using a reference angle from the wrong axis.

Correction: The reference angle is the acute angle between the terminal arm and the horizontal axis.

Lesson summary

The unit circle has radius $1$ and centre at the origin.
The point for angle $\theta$ is $(\cos\theta,\sin\theta)$ .
Use common reference angles to find coordinate sizes, then use the quadrant to set signs.
Angles that differ by a full turn are coterminal and have the same coordinates.
Check coordinates using $x^2+y^2=1$ .

Check your understanding

Question 1

What are the exact coordinates for

240^\circ

?

$(\frac{1}{2},-\frac{\sqrt{3}}{2})$
$(-\frac{1}{2},-\frac{\sqrt{3}}{2})$
$(-\frac{\sqrt{3}}{2},-\frac{1}{2})$
$(-\frac{1}{2},\frac{\sqrt{3}}{2})$

Show answer and explanation

(-\frac{\sqrt{3}}{2},-\frac{1}{2})

240^\circ

is in Quadrant III with reference angle

60^\circ

. The

60^\circ

coordinate sizes are

\frac{1}{2}

and

\frac{\sqrt{3}}{2}

, and both signs are negative.

Question 2

Which angle is coterminal with

-60^\circ

and lies between

0^\circ

and

360^\circ

?

$60^\circ$
$240^\circ$
$300^\circ$
$420^\circ$

Show answer and explanation

300^\circ

Adding one full turn gives

-60^\circ+360^\circ=300^\circ

. This angle is in the requested interval and has the same terminal arm.

Key terms

Unit circle: A circle with radius $1$ and centre at the origin.
Standard position: An angle with its vertex at the origin and its initial arm on the positive $x$ -axis.
Terminal arm: The ray that shows where an angle ends after turning.
Reference angle: The acute angle between a terminal arm and the horizontal axis.
Coterminal angles: Angles that end at the same point on the unit circle.
Exact value: A value written without rounding, such as $\frac{\sqrt{2}}{2}$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation TR2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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TR2 · Use the unit circle to determine coordinates and angle relationships

Alberta Grade 12 Mathematics

Alberta Mathematics 30-1 · TR2

What you will learn

1. Review: angles and coordinates

2. The unit circle and its coordinates

3. Signs, symmetry, and related angles

4. A reliable method

Reference coordinates and quadrant signs

Find coordinates in Quadrant II

Use a coterminal angle

Compare related angles

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Math 30-1

About this lesson and its review