SAT Math — radians, arc length and the unit circle

The College Board's description of the SAT circle skill includes two lines that many students meet for the first time in a precalculus class and then half-forget: solve problems using either radian measure or trigonometric ratios in the unit circle, and convert between angle measures in degrees and radians. The reference sheet in the testing app states that a circle has 360 degrees of arc and 2π radians of arc, but it does not say what to do with that fact. This material turns it into working habits.

The twelve quiz questions fall into three groups. The first is conversion in both directions, from 150 degrees to radians and from 7π/4 radians back to degrees, with distractors built from the usual slips of dividing by 90 or 360 instead of 180. The second is measurement with the angle already in radians, where the formulas are shorter than their degree versions: arc length is the radius times the angle, and a sector's area is half the radius squared times the angle. Students find an arc, a sector, an angle from an arc and a radius, a circumference from an arc and an angle, and the angle written as a multiple of π, the form a typed answer often takes. A rolling-wheel question applies the same idea to distance traveled.

The third group works on the unit circle: the coordinates of a point at 2π/3, the tangent of 5π/4, the angle that belongs to a given point, and what adding π does to a sine. Each one is solved the same way, by finding the reference angle and then taking the sign from the quadrant, and the explanations show that route every time rather than relying on a memorized table.

Seven questions carry a drawing: sectors and arcs with their radius and angle marked, and unit circles with the angle drawn from the positive x-axis. The drawings never label the value being asked for, and every question also states its numbers in words. Each explanation says what mistake leads to each wrong option, for example using the diameter where the radius belongs or reporting the sector's area when the arc was asked for.

The flashcards collect the two conversion factors, the arc-length and sector-area formulas in radians, the signs of sine, cosine and tangent in each quadrant, three special-angle values, and the identity for a half turn.

The material offers a quiz and flashcards. It is practice written by Zestly and is not affiliated with or endorsed by the College Board.

• Convert angle measures between degrees and radians
• Find arc lengths, sector areas, radii and central angles with the angle in radians
• Read sine, cosine and tangent of special angles from the unit circle using reference angles and quadrant signs
• Identify an angle in radians from a point on the unit circle
• Apply arc length to a rolling-wheel distance

Practice material written by Zestly, based on the College Board Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), Geometry and Trigonometry domain, skill Circles (radian measure, unit circle, degree-radian conversion).