Use the unit circle or special-angle reasoning to connect angles and coordinates (SAT). | Circles | SAT Math

On the unit circle, coordinates come from cosine and sine. Special angles and quadrant signs give the exact value.

Core Idea

On the unit circle, the point at angle $𝜃$ has coordinates $(c o s 𝜃, s i n 𝜃)$ . Special angles (30°, 45°, 60° and their multiples) give exact coordinate values.

Understanding

Rule: On the unit circle, coordinates are $(c o s 𝜃, s i n 𝜃)$ .

Start with the reference angle.
Use the exact special-angle coordinates.
Then fix the signs for the quadrant.

Cosine is the x-coordinate, sine is the y-coordinate.

Step by Step

Find the reference angle (the acute angle the terminal side makes with the x-axis).
Look up or recall the coordinates for that reference angle on the unit circle.
Adjust signs based on the quadrant: x is negative in Q2 and Q3, y is negative in Q3 and Q4.
Read off $c o s 𝜃 (𝑥 -$ coordinate) or $s i n 𝜃$ (y-coordinate) as needed.

Misconceptions

Swapping sine and cosine — cosine is the x-coordinate, sine is the y-coordinate.
Forgetting to adjust signs for the quadrant.
Using degree values in radian formulas (or vice versa) without converting.

Question

Worked Example

What is the value of $c o s 2 𝜋 3$ ?

Select an answer to see the explanation