The Unit Circle Is the Sine Wave
Stop memorizing a chart. Watch one point walk around a circle and draw the sine and cosine graphs on its own. Once you see it, it sticks.
A point on the unit circle at angle θ is (cos θ, sin θ).
Cosine is how far left or right the point is. Sine is how far up or down it is. That's it. Everything else on this page follows from that.
The unit circle is a circle with radius 1, centered at the origin. Start at the point (1, 0) on the right side and walk counterclockwise. The distance you have walked along the circle is the angle in radians. A full lap is 2π (about 6.28) because the circumference of a radius-1 circle is 2π.
1. Height becomes the sine wave
Press play. As the point goes around the circle, follow its height straight across to the right. Plot that height for every angle and you get the sine wave.
2. Width becomes the cosine wave
Cosine works the same way, but it tracks the point's left–right position. So we follow the point straight down and let the cosine wave hang below the circle. Then press “Stand it up” to rotate it into the cosine graph you see in class.
Notice that the cosine wave starts at the top (at angle 0 the point is all the way to the right, x = 1), while the sine wave starts in the middle (height 0). Cosine is the sine wave started a quarter-lap early: cos θ = sin(θ + π/2).
3. The 4 points that anchor both waves
The top, bottom, left and right of the circle are the peaks, valleys and zero-crossings of the waves. Learn these four and you can sketch either graph from memory.
| Angle | Point on circle | cos θ (x) | sin θ (y) | What the waves are doing |
|---|---|---|---|---|
| 0 (0°) | (1, 0) — far right | 1 | 0 | Cosine at its peak; sine crossing zero, going up |
| π/2 (90°) | (0, 1) — top | 0 | 1 | Sine at its peak; cosine crossing zero, going down |
| π (180°) | (−1, 0) — far left | −1 | 0 | Cosine at its lowest; sine crossing zero, going down |
| 3π/2 (270°) | (0, −1) — bottom | 0 | −1 | Sine at its lowest; cosine crossing zero, going up |
| 2π (360°) | (1, 0) — back to start | 1 | 0 | Both waves start over |
4. The full unit circle: tap any point
Here are all 16 special angles. Tap a point to see its exact values and exactly where it lands on the sine and cosine waves. To quiz yourself, turn the labels off and try to name each point before you tap it.
Tap any dot on the circle.
5. Four patterns, so there's nothing to memorize
Pattern 1: Count √1, √2, √3, all over 2
In the first quadrant, as the angle goes 30° → 45° → 60°, the sine values count up:
Cosine is the same list backwards (√3/2, √2/2, 1/2). That's because as the point climbs higher, it moves closer to the y-axis.
Pattern 2: The quadrant tells you the signs
Cosine is x and sine is y, so the signs are just the signs of x and y in each quadrant. Right half: cosine positive. Top half: sine positive.
Memory trick, starting in Q I and going counterclockwise: All Students Take Calculus (All, Sin, Tan, Cos).
Pattern 3: Every point is a mirror of a first-quadrant point
150°, 210° and 330° are all 30° away from the x-axis. They have the same numbers as 30° (√3/2 and 1/2), and only the signs change. That 30° is called the reference angle. Find the reference angle, take its values, and apply the quadrant's signs.
Pattern 4: Tangent is the slope of the radius
tan θ = sin θ / cos θ = rise / run. At 45° the radius has slope 1. At 90° it's vertical, so tangent is undefined, because you can't divide by cos 90° = 0.
6. Quick check: 8 questions
Use the circle in your head: cosine = x, sine = y. Each question shows you the point after you answer.
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