SAT Math · Geometry and Trigonometry

The unit circle

Q: How do you memorize the unit circle?

You only need three numbers: 1/2, √2/2, and √3/2. In the first quadrant, sine grows from 0 to 1 as the angle goes from 0° to 90°, and cosine shrinks from 1 to 0. Every other quadrant repeats those values with a sign change.

Q: Is the unit circle on the SAT?

Yes. Radians and trigonometric ratios on the unit circle are part of the Geometry and Trigonometry domain of SAT Math, which is about 15% of the Math section.

Q: Is the unit circle on the SAT reference sheet?

No. The reference sheet gives the special right triangles and says a circle has 360 degrees, or 2π radians, of arc. You can rebuild the unit circle values from those triangles.

Q: How do you convert degrees to radians?

Multiply the angle in degrees by π/180. For example, 60° × π/180 = π/3. To go back, multiply radians by 180/π.

Q: How can Aristotle help with SAT trigonometry?

Aristotle is a voice AI SAT tutor. Your child talks through a trig question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including radians and the unit circle.

Sine, cosine, and radians from one circle, with the values SAT Math expects you to know.

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Updated October 30, 2026

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

At $30^\circ$ , or $\tfrac{\pi}{6}$ , the point is $\left(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right)$ .

What is the unit circle?

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

A circle of radius 1 centered at the origin. Swing an arm of length 1 counterclockwise through an angle $\theta$ . Wherever its tip lands, the $x$ -coordinate is $\cos\theta$ and the $y$ -coordinate is $\sin\theta$ . That is the entire definition.

The tip of a radius-1 arm at angle θ is (cos θ, sin θ).

Why the legs are cos θ and sin θ

\cos\theta = \dfrac{x}{1} = x

\sin\theta = \dfrac{y}{1} = y

Drop a line from the point to the $x$ -axis: a right triangle with hypotenuse 1. SOH-CAH-TOA divides each leg by the hypotenuse, and dividing by 1 changes nothing. So the legs are not just related to cosine and sine; they are cosine and sine.

Hypotenuse 1, so the legs are exactly cos θ and sin θ.

Past 90°, only the signs change

135^\circ \;\to\; \left(-\tfrac{\sqrt{2}}{2},\ \tfrac{\sqrt{2}}{2}\right)

Every angle makes the same triangle as some first-quadrant angle, flipped across an axis. The side lengths repeat; the quadrant decides the signs. Learn the triangles for $30^\circ$ , $45^\circ$ and $60^\circ$ , and you know every special point.

The 135° triangle is the 45° triangle reflected across the y-axis.

Where the values come from

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Drop a line from the point straight down to the $x$ -axis. What shape do you get?
Answer
A right triangle with hypotenuse 1. Its legs are $\cos\theta$ across and $\sin\theta$ up.
Question 2
At $30^\circ$ , which special triangle is that, and what are its legs?
Answer
A 30-60-90 triangle with hypotenuse 1: legs $\tfrac{1}{2}$ and $\tfrac{\sqrt{3}}{2}$ , so the point is $\left(\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right)$ .
Question 3
Now go to $150^\circ$ . What changes?
Answer
Same size values, but in Quadrant II $x$ is negative: $\left(-\tfrac{\sqrt{3}}{2}, \tfrac{1}{2}\right)$ .

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

What is $150^\circ$ in radians?

Question 2

What is the value of $\cos\left(\tfrac{2\pi}{3}\right)$ ?

Question 3

In a right triangle, one acute angle measures $x^\circ$ , and $\sin(x^\circ) = 0.6$ . What is $\cos\big((90 - x)^\circ\big)$ ?

Question 4

A point on the unit circle lies in Quadrant IV and has an $x$ -coordinate of $\tfrac{3}{5}$ . What is its $y$ -coordinate?

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The key facts

A point on the circle

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

Cosine is how far across, sine is how far up.

Pythagorean identity

$\sin^2\theta + \cos^2\theta = 1$

Every point on the circle satisfies $x^2 + y^2 = 1$ , so this holds for every angle.

Degrees to radians

$180^\circ = \pi \text{ radians}$

Multiply degrees by $\tfrac{\pi}{180}$ to get radians. Multiply radians by $\tfrac{180}{\pi}$ to get degrees.

Arc length

$s = r\theta$

With $\theta$ in radians. On the unit circle $r = 1$ , so the arc length equals the angle.

First-quadrant values to know
Degrees	Radians	$\cos\theta$ ( $x$ )	$\sin\theta$ ( $y$ )
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\tfrac{\pi}{6}$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{2}$
$45^\circ$	$\tfrac{\pi}{4}$	$\tfrac{\sqrt{2}}{2}$	$\tfrac{\sqrt{2}}{2}$
$60^\circ$	$\tfrac{\pi}{3}$	$\tfrac{1}{2}$	$\tfrac{\sqrt{3}}{2}$
$90^\circ$	$\tfrac{\pi}{2}$	$0$	$1$

Try it

Walk around the unit circle

Step through the special angles and watch the point and signs change.

Angle θ45° = π/4

Point(√2/2, √2/2), in quadrant I

cos θ√2/2

sin θ√2/2

tan θ1

Common mistakes

Swapping sine and cosine

Cosine is the $x$ -coordinate and sine is the $y$ -coordinate. Alphabetical order helps: $(x, y)$ goes with $(\cos, \sin)$ .

Leaving the calculator in the wrong mode

Radians in, degree mode on: every answer is wrong. Check the setting first.

Forgetting the sign past 90°

Find the value from the reference angle, then decide the sign from the quadrant. $\cos 120^\circ$ is $-\tfrac{1}{2}$ , not $\tfrac{1}{2}$ .

Flipping the conversion factor

Degrees to radians is $\times \tfrac{\pi}{180}$ . If your answer for $90^\circ$ is not $\tfrac{\pi}{2}$ , the fraction is upside down.

How it shows up on the SAT

Geometry and Trigonometry is about 15% of SAT Math. The unit circle is not on the reference sheet, but the special triangles are. Expect radians, special-angle values, and sin θ = cos(90° − θ).

What Aristotle tracks in circles

7 SAT Math skills, including:

Solving problems using circle properties
Using radians and trigonometric ratios in the unit circle
Writing equations for circles in the xy-plane
Relating circle equations to circle graphs

What is the unit circle?

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. The point at angle θ on the circle has coordinates (cos θ, sin θ), which is how it defines sine and cosine for every angle.

How do you memorize the unit circle?

Is the unit circle on the SAT?

Is the unit circle on the SAT reference sheet?

How do you convert degrees to radians?

How can Aristotle help with SAT trigonometry?

More SAT concepts

The Pythagorean theorem

a² + b² = c² for any right triangle, plus the triples and special triangles that save time on SAT Math.

Equation of a circle

Read the center and radius from (x − h)² + (y − k)² = r², and complete the square when the equation is expanded.

Volume of a cone and a pyramid

Why both formulas have a one third in them, and how to use them on SAT Math.

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