AP subjects/AP Precalculus/Interactive Unit Circle
CED 3.2AP Precalculus

Interactive Unit Circle

Use this free interactive unit circle to see where sine, cosine and tangent come from: drag a point around the circle and watch its coordinates, cos⁡θ\cos\theta and sin⁡θ\sin\theta, update in real time. It is built for AP Precalculus Topic 3.2, where the unit circle is the definition of the three trig functions.

Controls
anglestepsnapdrag

How to use the simulator

The circle has radius 1 and is centered at the origin. A navy terminal ray runs from the origin to a white point on the circle, and a gold arc marks the angle θ\theta, measured counterclockwise from the positive x-axis. The teal segment along the x-axis is cos⁡θ\cos\theta (the horizontal distance), and the coral vertical segment is sin⁡θ\sin\theta (the height of the point).
  • Drag on the circle: press anywhere on the plot and drag; the point jumps to the angle in that direction.
  • angle θ slider: sets θ\theta from 0° to 360° in 1° steps. The value appears to the right of the slider.
  • −15° and +15° buttons: step the angle by 15°, which lands you on every special angle that is a multiple of 15°. Stepping past 360° wraps back to 0°.
  • 0°, 30°, 45°, 90° buttons: jump straight to those benchmark angles.
The readout panel shows the angle in degrees and in radians (rounded to two decimals, so 30° shows as 0.52 rad), then the point (x, y), cosθ, sinθ and tanθ to three decimals, and the quadrant (I, II, III, IV, or "on axis" at 0°, 90°, 180° and 270°). At 90° and 270° the tanθ row reads undefined.
Because the readouts are decimals, use them to check exact values you work out by hand: 0.8660.866 is 32\tfrac{\sqrt3}{2}, 0.7070.707 is 22\tfrac{\sqrt2}{2}, and 0.5770.577 is 33\tfrac{\sqrt3}{3}.

The formulas

For an angle θ\theta in standard position, the terminal ray meets the unit circle at the point (x,y)=(cos⁡θ,sin⁡θ),tan⁡θ=sin⁡θcos⁡θ=yx  (x≠0).(x, y) = (\cos\theta, \sin\theta), \qquad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}\ \ (x \neq 0). Tangent is also the slope of the terminal ray, which is why it is undefined when the ray is vertical.
  • Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, because every point on the circle is 1 unit from the origin.
  • Degree–radian conversion: 180∘=π180^\circ = \pi radians, so multiply degrees by π180\tfrac{\pi}{180} to get radians and radians by 180π\tfrac{180}{\pi} to get degrees. For example, 150∘=5π6150^\circ = \tfrac{5\pi}{6}.
  • Reference angle: the acute angle between the terminal ray and the x-axis. In Quadrant II it is 180∘−θ180^\circ - \theta, in III it is θ−180∘\theta - 180^\circ, in IV it is 360∘−θ360^\circ - \theta.
  • Signs by quadrant: I: all positive. II: sine positive. III: tangent positive. IV: cosine positive.
  • Periodicity: adding 360∘360^\circ (2π2\pi) returns to the same point, so sine and cosine repeat every 2π2\pi; tangent repeats every π\pi.
The special values to know: sin⁡30∘=12\sin 30^\circ = \tfrac12, cos⁡30∘=32\cos 30^\circ = \tfrac{\sqrt3}{2}; sin⁡45∘=cos⁡45∘=22\sin 45^\circ = \cos 45^\circ = \tfrac{\sqrt2}{2}; sin⁡60∘=32\sin 60^\circ = \tfrac{\sqrt3}{2}, cos⁡60∘=12\cos 60^\circ = \tfrac12. Every other special angle is one of these with a sign attached.

Worked example

Find sin⁡θ\sin\theta, cos⁡θ\cos\theta and tan⁡θ\tan\theta for θ=5π6\theta = \tfrac{5\pi}{6}.
Step 1: convert and locate. 5π6⋅180∘π=150∘\tfrac{5\pi}{6}\cdot\tfrac{180^\circ}{\pi} = 150^\circ. That is between 90° and 180°, so the angle is in Quadrant II.
Step 2: reference angle. 180∘−150∘=30∘180^\circ - 150^\circ = 30^\circ (or π−5π6=π6\pi - \tfrac{5\pi}{6} = \tfrac{\pi}{6}). So the values have the same size as the 30° values: 32\tfrac{\sqrt3}{2} and 12\tfrac12.
Step 3: attach signs. In Quadrant II, x is negative and y is positive: cos⁡5π6=−32,sin⁡5π6=12,tan⁡5π6=1/2−3/2=−13=−33.\cos\tfrac{5\pi}{6} = -\tfrac{\sqrt3}{2},\qquad \sin\tfrac{5\pi}{6} = \tfrac12,\qquad \tan\tfrac{5\pi}{6} = \frac{1/2}{-\sqrt3/2} = -\tfrac{1}{\sqrt3} = -\tfrac{\sqrt3}{3}.
Check in the simulator: press 90°, then +15° four times. The readout shows 150° (2.62 rad), point (−0.866, 0.500), tanθ −0.577, quadrant II.
Follow-up: 7π6\tfrac{7\pi}{6} (210°). Same 30° reference angle, but now in Quadrant III where both coordinates are negative: cos⁡7π6=−32\cos\tfrac{7\pi}{6} = -\tfrac{\sqrt3}{2}, sin⁡7π6=−12\sin\tfrac{7\pi}{6} = -\tfrac12, and tan⁡7π6=33\tan\tfrac{7\pi}{6} = \tfrac{\sqrt3}{3} (positive, since negative over negative). Set the slider to 210° and you should see (−0.866, −0.500) and tanθ 0.577. The point is the reflection of the 30° point through the origin.

Common mistakes on the AP exam

  • Swapping sine and cosine. Cosine is the x-coordinate, sine is the y-coordinate. Alphabetical order helps: (x, y) matches (cos, sin).
  • Dropping the sign. Students find the right reference value and forget the quadrant. Locate the quadrant first, then fill in the magnitude.
  • Wrong reference angle. The reference angle is always measured to the x-axis, never the y-axis. For 120° it is 60°, not 30°.
  • Calculator in the wrong mode. When a calculator is allowed, a radian input in degree mode (or the reverse) gives a confidently wrong answer.
  • Calling tangent zero instead of undefined. At 90° and 270°, cos⁡θ=0\cos\theta = 0, so tan⁡θ\tan\theta is undefined; at 0° and 180° it is sin⁡θ\sin\theta that is zero, so tan⁡θ=0\tan\theta = 0.
  • Mixing up 32\tfrac{\sqrt3}{2} and 12\tfrac12. For 30° the point is closer to the x-axis, so the y-value (sine) is the smaller one, 12\tfrac12.

When the AP exam uses this

Unit circle values underpin most of Unit 3: evaluating sinusoidal functions, finding where a function equals zero or reaches a maximum, solving trigonometric equations on an interval, and identifying asymptotes of tangent. Even on sections where a graphing calculator is permitted, you are expected to reason out exact values like sin⁡5π6\sin\tfrac{5\pi}{6} from the unit circle rather than rely on decimal output.
Embed this simulator on your class page

Free for classroom use. Paste this into your site, LMS page or blog; keep the credit link under it.

More AP Precalculus simulators

CED 1.12
Function Transformations Visualizer
CED 3.14
Polar Function Grapher
CED 2.10
Exponential vs Logarithmic Functions Explorer
CED 1.8
Rational Function Asymptotes Explorer
CED 1.6
Polynomial End Behavior Explorer
CED 3.6
Sinusoidal Function Model Builder
CED 4.1
Parametric Equations Grapher
CED 4.8
Vector Addition & Operations Visualizer