Knowledge Notes | Carey AP Precalculus Practice Lab

Reference-angle magnitudes, quadrant signs and nonunit-circle radius scaling are separated into three steps. — For a circle centered at the origin with radius $r>0$ , use $(x,y)=(r\cos\theta,r\sin\theta)$ . Determine each sign from the actual quadrant. For nonquadrantal angles use the acute reference angle; handle axis angles directly. The paper shapes are illustrative.

What a strong response shows

Start with the unit-circle magnitudes for a familiar reference angle.
Choose signs from the actual terminal quadrant, rather than copying the reference-angle signs.
Use $(r\cos\theta,r\sin\theta)$ ; both components scale by the radius.

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What You Need to Know

Key idea

Points on a circle centered at the origin are controlled by radius and angle. On the unit circle, the coordinates are sine and cosine values.

Fast decision rule

Use the radius and angle together; do not treat every circle as if its radius were $1$ .

Worked steps

Identify the radius, locate the terminal angle, and scale the unit-circle coordinates by that radius.

Common mistake

Reading the unit-circle coordinate directly when the given circle has another radius.

Mini check

What changes when the circle radius changes from $1$ to $r$ ?

Learning Objectives

Key idea

3.3.A — Determine coordinates of points on a circle centered at the origin.

Fast decision rule

For a coordinate prompt, find the reference angle and quadrant before choosing signs.

Worked steps

$1$ . Locate the angle. $2$ . Find the first-quadrant coordinate pattern. $3$ . Apply quadrant signs. $4$ . Multiply by the radius.

Common mistake

Applying a positive radius as if it could change a coordinate sign.

Mini check

Which step handles the radius?

Key Concepts

Key idea

Reference angles give magnitudes; symmetry gives the signs and coordinate order.

Fast decision rule

Magnitude first, quadrant second, radius last.

Worked steps

Reflect a first-quadrant point across an axis to obtain the matching coordinate magnitude, then change the sign of the reflected coordinate.

Common mistake

Reflecting across the wrong axis and swapping sine with cosine.

Mini check

Which reflection changes the sign of the x-coordinate?

Relevant Representations

Key idea

A circle, a right triangle, and a coordinate table are equivalent ways to organize the same point.

Coordinate sign guide
Quadrant	x-coordinate	y-coordinate
$I$	positive	positive
II	negative	positive
III	negative	negative
IV	positive	negative

Fast decision rule

Use the circle for location, the triangle for magnitude, and the table for repeated exact values.

Worked steps

Build a reference triangle, label the horizontal and vertical legs, then transfer the signs to the circle.

Common mistake

Labeling the opposite and adjacent legs from the page instead of from the chosen angle.

Mini check

For a terminal ray that is not on an axis, which acute angle with the horizontal axis is its reference angle? Handle axis angles directly with the circle coordinates.

Essential Relationships or Formulas

Key idea

Circle coordinates scale the unit-circle values by the radius.

x=r\cos\theta

y=r\sin\theta

x^2+y^2=r^2

Fast decision rule

Use the radius in both coordinates and keep the angle attached to the correct coordinate.

Worked steps

Substitute the radius and exact sine or cosine value, then simplify the radical or fraction.

Common mistake

Using the radius only once or switching sine and cosine.

Mini check

What equation should the final coordinates satisfy?

Worked Examples

Key idea

A coordinate can be found without a calculator when the angle has a familiar reference triangle.

Fast decision rule

Match the reference angle to a known triangle, then apply the circle’s radius and quadrant.

Worked steps

For a radius-2 circle in Quadrant IV with a 30-degree reference angle, use the $30-60-90$ magnitudes, make $y$ negative, and check the radius equation.

Common mistake

Making both coordinates negative in Quadrant IV.

Mini check

How can the radius equation verify the result?

Common Mistakes

Key idea

Coordinate errors usually come from wrong quadrant, swapped coordinates, or missing radius.

Fast decision rule

Use the order $x$ then $y$ , and sign each coordinate from the quadrant.

Worked steps

Read the angle counterclockwise, mark the quadrant, label $x$ and $y$ separately, and substitute.

Common mistake

Writing a coordinate pair in y,x order.

Mini check

Which coordinate is listed first in an ordered pair?

AP Exam Tips

Key idea

Always include a quick circle check when a coordinate is requested.

Practice

Use the Questionbank set below to test this skill.

Fast decision rule

Verify quadrant signs and the radius equation before submitting.

Worked steps

Show the exact pair, state the quadrant, and use the circle relationship as evidence.

Common mistake

Rounding before the radius check.

Mini check

Does your point land on the stated circle?

TRY IT IN A NEW SETTING

Can you transfer the method?

Find the point at $\theta=4\pi/3$ on a circle of radius $6$ . Explain how a unit-circle answer changes when the radius changes.

Compare your reasoning

The unit-circle point is $(-1/2,-\sqrt3/2)$ . Scaling both coordinates by $6$ gives $(-3,-3\sqrt3)$ . The angle and signs stay the same; coordinate magnitudes scale.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

3.3 · Sine and Cosine Function Values

Topic 3.3 · Sine and Cosine Function Values

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?