Unit Circle on Desmos: SAT Radian Problems Solved

On Desmos, you can check any unit-circle value by typing cos(7π/6) or sin(7π/6) in radian mode, so you don't need to memorize the circle. For angles bigger than one turn, mod(angle, 2π) gives the matching angle between 0 and 2π. This guide shows both moves on SAT-style radian problems, with a by-hand check for each.

You can type every line below into the College Board version of Desmos on our Desmos on the SAT page. You get the rule, two worked examples, one guided example and five practice problems with answers.

The rule: a point on the unit circle is (cos θ, sin θ)

The unit circle has its center at the origin and a radius of 1. Start at (1, 0) and turn counterclockwise through an angle θ. You land on the point (cos θ, sin θ). A clockwise turn gives a negative angle.

• Co-terminal angles. Two angles that differ by a multiple of 2π land on the same point. 23π and π are co-terminal, because 23π = π + 11(2π).
• Degrees to radians. Multiply by π/180. 45° is π/4, and 300° is 5π/3.
• Radians to degrees. Multiply by 180/π. π/3 is 60°.

These are the angles used on this page:

Check the angle mode first

Desmos can work in radians or in degrees, and the same keystrokes give different answers in each mode. The Desmos Help Center says its standard graphing calculator starts in radians, its scientific calculator starts in degrees, and testing versions of the calculator often start in degrees. We have not confirmed which mode Bluebook's calculator opens in, so check it yourself before you type a trig value.

The check takes one line:

1  cos(π)

If Desmos shows -1, you are in radians. If it shows about 0.9985, you are in degrees, because Desmos took π to mean about 3.14 degrees. On desmos.com, you switch modes in the Graph Settings menu. To type π, type the letters pi.

Worked example 1 (I do): which angle lands on (-1, 0)?

SAT-style question. In the xy-plane, point H(-1, 0) lies on the unit circle, which has its center at the origin O. An angle in standard position has its terminal side passing through H. Which of the following could be the measure of this angle, in radians?

A) 25π/2   B) 27π/2   C) 22π   D) 23π

Step 1. Name the target. H = (-1, 0) sits at π, halfway around the circle. So the answer is the choice that is co-terminal with π.

Step 2. Reduce all four choices at once. Desmos accepts a list anywhere it accepts a number, so put the choices in square brackets. Dividing by π makes the output easy to read:

1  mod([25π/2, 27π/2, 22π, 23π], 2π)/π

The output is 0.5, 1.5, 0 and 1. Those numbers mean π/2, 3π/2, 0 and π.

Step 3. Pick the match. Only 23π reduces to π. Confirm with the coordinates:

2  cos(23π)
3  sin(23π)

Line 2 gives -1. If line 3 shows a tiny number written with ×10^-15, read it as 0. That is rounding inside the calculator. The point is (-1, 0), which is H.

By-hand check. 23π = 22π + π = 11(2π) + π, so 23π is eleven full turns plus a half turn. The other choices fail: 25π/2 = 6(2π) + π/2 lands on (0, 1), 27π/2 = 6(2π) + 3π/2 lands on (0, -1), and 22π = 11(2π) lands on (1, 0). The answer is D.

Worked example 2 (I do): rotate a point

SAT-style question. Point A(0, 1) lies on the unit circle centered at the origin. Point P is the image of A after a counterclockwise rotation of 7π/6 radians about the origin. What are the coordinates of P?

Step 1. Find A's angle. (0, 1) is at the top of the circle, so A sits at π/2.

Step 2. Add the rotation. π/2 + 7π/6 = 3π/6 + 7π/6 = 10π/6 = 5π/3. Counterclockwise means you add. A clockwise turn would mean you subtract.

Step 3. Evaluate on Desmos. Type the sum inside cos and sin so you don't have to simplify first:

1  cos(π/2 + 7π/6)
2  sin(π/2 + 7π/6)

Line 1 gives 0.5 and line 2 gives about -0.866. The answer choices will use exact values, so match the decimals: 0.5 is 1/2, and -0.866 is -√3/2 (√3/2 is about 0.866).

To see the rotation, add the circle and the point:

3  x^2 + y^2 = 1
4  (cos(5π/3), sin(5π/3))

The point sits in the fourth quadrant, right of the y-axis and below the x-axis.

By-hand check. 5π/3 is 300°, which is 60° short of a full turn. The 60° reference angle gives the values 1/2 and √3/2. In the fourth quadrant, x is positive and y is negative. So P = (1/2, -√3/2).

Guided example (we do): co-terminal with 3π/4

SAT-style question. Which of the following angles is co-terminal with 3π/4?

A) -3π/4   B) 11π/4   C) π/4   D) 5π/4

Step 1. The target point is 3π/4, which is (-√2/2, √2/2), or about (-0.7071, 0.7071).

Step 2. Evaluate cos for all four choices with one list:

1  cos([-3π/4, 11π/4, π/4, 5π/4])

The output is -0.7071, -0.7071, 0.7071 and -0.7071. Three choices share the cosine of 3π/4, so cosine alone can't decide this question. Two different points on the circle can have the same x-coordinate.

Step 3 (your step). Type sin([-3π/4, 11π/4, π/4, 5π/4]). Find the choice whose sine is also 0.7071, then confirm it by hand: its difference from 3π/4 must be a whole number of 2π turns.

Check your answer: the sines are -0.7071, 0.7071, 0.7071 and -0.7071. Only 11π/4 matches both values. By hand, 11π/4 - 3π/4 = 8π/4 = 2π, one full turn. The answer is B. Choice C, π/4, matches the sine only, and choices A and D match the cosine only.

Want more questions like this one? Work through the free Radian Measure lesson notes.

Your turn (you do): 5 practice problems

1. Which angle is co-terminal with π/3? A) 4π/3  B) -5π/3  C) 2π/3  D) -π/3
2. What are the coordinates of the point on the unit circle at 7π/4?
3. Convert 150° to radians.
4. Point P is at (-1, 0) on the unit circle. It rotates counterclockwise by 9π/2 radians about the origin. What are its new coordinates?
5. The angle θ is between π and 3π/2, and sin θ = -1/2. What is θ?

Answers

1. B. -5π/3 + 2π = π/3. The other choices differ from π/3 by π, π/3 and 2π/3, none of them a full turn.
2. (√2/2, -√2/2). 7π/4 is 315°, in the fourth quadrant with a 45° reference angle. Desmos gives about 0.7071 and -0.7071.
3. 5π/6. 150 × π/180 = 5π/6.
4. (0, -1). P starts at π. π + 9π/2 = 11π/2, and mod(11π/2, 2π) = 3π/2, the bottom of the circle.
5. 7π/6. In the third quadrant sine is negative, and a 30° reference angle gives 1/2. 180° + 30° = 210°, which is 7π/6.

When algebra is faster

College Board's calculator guidance says some Math questions are better done without a calculator, even though one is allowed. Converting degrees to radians is one of them: 150 × π/180 takes one line on paper. Use Desmos when the angle is large, such as 23π, or when you need to test four answer choices at once.

Mistakes to avoid

• Working in the wrong mode. In degree mode, cos(7π/6) treats 7π/6 as about 3.7 degrees and returns a number near 1. Run the cos(π) check first.
• Checking only cosine. In the guided example, three choices matched cos(3π/4). Check sine too, or reduce the angle with mod.
• Misreading the decimal. -0.7071 is -√2/2 and -0.866 is -√3/2. Match the decimal to the exact form before you pick a choice.

Practice next

Start with the free Radian Measure lesson notes. For more questions, browse the SAT question bank.

Related guides: for sine, cosine and tangent in a triangle, read SAT trig ratios on Desmos. For questions that simplify sin and cos expressions, see SAT trig expressions on Desmos. For circle equations and tangent lines, try circle tangent slope with a Desmos check.

Checked on September 26, 2026 against College Board's SAT Suite calculator policy, the College Board version of the Desmos graphing calculator, and the Desmos Help Center FAQs and its articles on trigonometry, testing calculators and supported functions. Every answer on this page was recomputed before publishing. The Test Advantage is independent of College Board and Desmos.