Knowledge Notes | Carey AP Precalculus Practice Lab

The terminal point on a unit circle supplies signed horizontal and vertical coordinates, with tangent as their quotient. — On a unit circle, the terminal point is $(\cos\theta,\sin\theta)$ . Tangent is vertical divided by horizontal and is undefined when the horizontal coordinate is zero. The unnumbered sketch is conceptual.

What a strong response shows

Identify the angle's actual terminal point before using a reflected point.
For radius $r$ , sine is vertical coordinate/r, cosine is horizontal coordinate/r, and tangent is vertical/horizontal.
Check the quadrant signs and whether the required denominator is zero.

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What You Need to Know

Key idea

On the unit circle, an angle identifies a point. The point’s horizontal coordinate gives cosine, vertical coordinate gives sine, and tangent compares the two when defined.

Fast decision rule

Locate the terminal point first; read horizontal, vertical, then vertical-over-horizontal.

Worked steps

Convert the angle to a standard position, locate its quadrant, read the point, and use the sign from that quadrant.

Common mistake

Assigning signs from the reference angle without checking the quadrant.

Mini check

Which coordinate is cosine on the unit circle?

Learning Objectives

Key idea

3.2.A — Determine the sine, cosine, and tangent of an angle using the unit circle.

Fast decision rule

For a unit-circle prompt, use the terminal point; for a triangle prompt, use opposite, adjacent, and hypotenuse.

Worked steps

$1$ . Place the angle in standard position. $2$ . Find the terminal point. $3$ . Read cosine and sine coordinates. $4$ . Divide sine by cosine only when cosine is nonzero.

Common mistake

Using a right-triangle ratio before checking whether the angle is in a non-right-triangle setting.

Mini check

When is tangent undefined?

Key Concepts

Key idea

The unit circle has radius $1$ , so its coordinates are exactly the sine and cosine values of the angle.

Fast decision rule

Quadrant signs come from the coordinate signs, not from the size of the reference angle.

Worked steps

A point in Quadrant II has negative horizontal coordinate and positive vertical coordinate, so cosine is negative and sine is positive.

Common mistake

Thinking all trig values are positive because lengths in a triangle are positive.

Mini check

What are the signs of sine and cosine in Quadrant IV?

Relevant Representations

Key idea

The same angle can be shown by a ray, a unit-circle point, a right triangle, or a coordinate pair.

Unit-circle roles
Quantity	Geometric cue	Value read
cosine	horizontal projection	x-coordinate
sine	vertical projection	y-coordinate
tangent	vertical divided by horizontal	sine over cosine

Fast decision rule

Choose the representation that exposes the requested value most directly.

Worked steps

Draw the terminal ray, drop a perpendicular to the x-axis, and read the two coordinate projections.

Common mistake

Mixing degree and radian labels on the same diagram.

Mini check

Which representation makes quadrant signs immediate?

Essential Relationships or Formulas

Key idea

Unit-circle definitions preserve the same angle relationships across representations.

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

\tan\theta=\frac{\sin\theta}{\cos\theta}

x^2+y^2=1

Fast decision rule

Read coordinates first; simplify only after the sign and denominator are known.

Worked steps

Substitute the coordinate values into the requested ratio, keep radicals exact, and rationalize only if the response format requires it.

Common mistake

Dividing by a zero cosine value and reporting a finite tangent.

Mini check

What identity confirms that a point lies on the unit circle?

Worked Examples

Key idea

Exact trig values are quickest when the angle is a special unit-circle angle.

Fast decision rule

Find the reference angle, use the first-quadrant value, then apply quadrant signs.

Worked steps

For a Quadrant III angle with a 45-degree reference angle, use equal-magnitude coordinates and make both coordinates negative.

Common mistake

Changing the reference value instead of changing only its signs.

Mini check

Which part of the process determines the sign?

Common Mistakes

Key idea

Most errors are angle-format, quadrant, or denominator errors.

Fast decision rule

Check units, quadrant, and whether a denominator is zero in that order.

Worked steps

Rewrite the angle in the requested measure, sketch the quadrant, and inspect the coordinate used in a ratio.

Common mistake

Using a calculator decimal when an exact value is requested.

Mini check

What should remain exact in a no-calculator response?

AP Exam Tips

Key idea

A small unit-circle sketch can replace several lines of prose.

Practice

Use the Questionbank set below to test this skill.

Fast decision rule

State the quadrant and show the coordinate or ratio that supports the answer.

Worked steps

Name the terminal point, identify the sign, then give the exact value.

Common mistake

Giving only a number with no visible reasoning.

Mini check

Can a reader tell how the quadrant supports your sign?

TRY IT IN A NEW SETTING

Can you transfer the method?

A terminal point is $(-5,12)$ on a circle centered at the origin. Find the radius, sine, cosine and tangent, and justify their signs.

Compare your reasoning

The radius is $\sqrt{(-5)^2+12^2}=13$ . Thus $\sin\theta=12/13$ , $\cos\theta=-5/13$ , and $\tan\theta=-12/5$ . The point is in Quadrant II, so sine is positive and $\frac{\text{cosine}}{\text{tangent}}$ are negative.

Use it in a real question

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

Practise this topic Check the whole unit

3.2 · Sine, Cosine, and Tangent

Topic 3.2 · Sine, Cosine, and Tangent

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?