High school · Trigonometry
The unit circle in motion
Rotate a radius and watch its horizontal and vertical projections become cosine and sine.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
Understand what you are seeing
The idea behind the motion.
A point one unit from the origin has coordinates (cos θ, sin θ). Its two perpendicular projections form a right triangle, but the coordinates keep their signs as the point moves through all four quadrants. This extends trigonometry beyond acute triangles.
A relationship to keep
x = cos θ, y = sin θ; x² + y² = 1
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Read the radius
The navy segment always has length one. Positive angles move counterclockwise from the positive horizontal axis; a negative starting angle begins clockwise from that axis.
- STEP 2
Follow the two projections
The teal horizontal segment represents cosine and the amber vertical segment represents sine. Their lengths show magnitudes; the point's side of each axis determines the signs.
- STEP 3
Check a full revolution
Scrub through one turn. The point returns to its starting location after 360° or 2π radians, so sine and cosine repeat even while the accumulated angle keeps increasing.
Your turn to explain
Make a prediction. Test your reasoning.
At 150°, which coordinate is positive, and what are the exact coordinates?
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
Sine is positive and cosine is negative. The point is (−√3/2, 1/2), and its squared coordinates add to 3/4 + 1/4 = 1.
Work through a full lesson
Connect the animation to a worked example and practice questions.