The unit circle is tested more heavily on the ACT than on the SAT. You need to know the coordinates at standard angles and how signs change across quadrants.
The Setup
The unit circle has radius 1 centered at the origin. A point at angle has coordinates:
So is the x-coordinate, and is the y-coordinate.
The Key Angles
You only need to memorize 5 values in the first quadrant. Everything else is derived from these:
| Angle | ||
|---|---|---|
Memory trick: For sine, the numerators go — all divided by 2. For cosine, it is the same sequence reversed.
The Quadrant Rule: ASTC
Which trig functions are positive in each quadrant?
All - Sin - Tan - Cos (going counterclockwise from Quadrant I)
- Quadrant I (-): ALL positive
- Quadrant II (-): only SIN positive
- Quadrant III (-): only TAN positive
- Quadrant IV (-): only COS positive
Using Reference Angles
For any angle outside the first quadrant:
- Find the reference angle (acute angle to the nearest x-axis)
- Look up the trig value for that reference angle
- Apply the sign from the quadrant rule
Example:
- Reference angle:
- Quadrant II: sin is positive
Example:
- Reference angle:
- Quadrant III: cos is negative
Common Mistakes to Avoid
- Confusing sin and cos at and : At , (smaller) and (bigger). At , they swap. Rule: bigger angle = bigger sine.
- Wrong quadrant signs: Draw a quick coordinate plane if unsure. The sign of matches the sign of , and the sign of matches the sign of .
- Degrees vs radians: , , . The ACT uses both.
ACT Pro Tip
The ACT loves asking for trig values at angles like , , or . The process is always the same: find the reference angle, look up the first-quadrant value, apply the sign. Practice this process 5 times and it becomes automatic — under 15 seconds per question.
Master the unit circle with our Unit Circle lesson — ACT-exclusive content.