How can I use reference angles to locate an angle’s unit-circle coordinates?

Separate the task into two parts: the reference angle helps you get the coordinate values, and the original angle tells you the signs. A unit-circle point is written as (cos θ, sin θ), so cosine is the x-coordinate and sine is the y-coordinate. The reference angle is the smallest angle between the terminal side and the horizontal axis; it is not usually the original angle. [S1]

Try this sequence:

1. Identify the original angle’s quadrant.
2. Find the reference angle.
3. Use the reference angle’s first-quadrant coordinate values.
4. Apply the signs for the original quadrant.

With x listed first, the sign patterns are Quadrant I (+,+), II (−,+), III (−,−), and IV (+,−). The original angle determines which quadrant—and therefore which signs—to use.

The original angle, 150°, lies in Quadrant II. The angle between its terminal side and the horizontal axis is 180° − 150° = 30°, so 30° is its reference angle. The first-quadrant point for 30° has coordinate values (√3/2, 1/2). For the original angle, the point is in Quadrant II: x is negative and y is positive. So the coordinates for 150° are (−√3/2, 1/2). [S1]

The reference angle is not a replacement for the original angle. It gives you familiar coordinate values; the original angle gives you the quadrant and signs. As a check, the signs in your answer should match Quadrant II.

Use the same idea and keep the angle in radians throughout. For 7π/6, the terminal side is in Quadrant III. Subtract π to find the reference angle: 7π/6 − π = π/6. The first-quadrant coordinate values for π/6 are (√3/2, 1/2). Since the original angle is in Quadrant III, both coordinates are negative. The point for 7π/6 is therefore (−√3/2, −1/2). [S1]

The reference angle π/6 supplies the familiar values, but the original angle 7π/6 determines that the point belongs in Quadrant III. That distinction is the same in degrees and radians.

Yes. Write down the original angle first, because it tells you where the point lies. Then record the reference angle and its first-quadrant coordinate values. Apply signs using the original quadrant, keeping x first and y second:

Original angle: ____ Quadrant: ____ Reference angle: ____ First-quadrant coordinate values: (____, ____) Signs for x and y from the original quadrant: ____, ____ Final point: (____, ____)

For example, with 150°, the original angle is in Quadrant II, the reference angle is 30°, and the first-quadrant values are (√3/2, 1/2). Assigning the Quadrant II signs gives (−√3/2, 1/2). Check that you used the reference angle for the values and the original angle for the signs. [S1]