Unit Circle Calculator: The Trigonometry Reference You Actually Need
The unit circle is a circle with radius 1, centered at the origin. For any angle θ, the coordinates of the point on the circle are (cos θ, sin θ). This single fact encodes all of trigonometry.
Key Angles and Their Values
| Angle | Degrees | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0° | 0 | 1 | 0 |
| π/6 | 30° | 1/2 | √3/2 | 1/√3 |
| π/4 | 45° | √2/2 | √2/2 | 1 |
| π/3 | 60° | √3/2 | 1/2 | √3 |
| π/2 | 90° | 1 | 0 | undefined |
| π | 180° | 0 | -1 | 0 |
| 3π/2 | 270° | -1 | 0 | undefined |
| 2π | 360° | 0 | 1 | 0 |
The Memory Trick for 30-45-60
Sin values for 0°, 30°, 45°, 60°, 90°: √0/2, √1/2, √2/2, √3/2, √4/2
= 0, 1/2, √2/2, √3/2, 1
Cosine is sin in reverse: 1, √3/2, √2/2, 1/2, 0
Signs by Quadrant
The CAST rule (or "All Students Take Calculus"):
- Quadrant I (0–90°): All positive
- Quadrant II (90–180°): Sine positive
- Quadrant III (180–270°): Tangent positive
- Quadrant IV (270–360°): Cosine positive
Reference Angles
For any angle outside 0–90°, find the reference angle (acute angle to the x-axis), compute the trig value, then apply the correct sign for the quadrant.
sin(150°) = sin(30°) = 1/2 (Quadrant II, sine positive) ✓
cos(240°) = -cos(60°) = -1/2 (Quadrant III, cosine negative) ✓
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Frequently Asked Questions
How do I find sin(150°) without memorizing every angle?
Use the reference angle method: 150° is in Quadrant II, so find the acute angle to the x-axis, which is 30°. Then compute sin(30°) = 1/2, and apply the Quadrant II rule (sine positive), giving sin(150°) = 1/2. This works for any angle outside 0–90°.
Why do I need to know the unit circle if I can just use a calculator?
A calculator gives you decimal approximations, but the unit circle shows you exact values like √3/2 or 1/√3, which are essential for algebra, calculus, and physics. Understanding the unit circle also helps you recognize patterns and catch errors in your work.
What's the memory trick for sin and cos of 30°, 45°, and 60°?
For sine, use √0/2, √1/2, √2/2, √3/2, √4/2 for angles 0°, 30°, 45°, 60°, 90° respectively—this simplifies to 0, 1/2, √2/2, √3/2, 1. For cosine, simply reverse the list: 1, √3/2, √2/2, 1/2, 0.
How do I know if sin or cos should be positive or negative?
Use the CAST rule: in Quadrant I all trig functions are positive; Quadrant II only sine is positive; Quadrant III only tangent is positive; Quadrant IV only cosine is positive. For example, cos(240°) is in Quadrant III where cosine is negative, so cos(240°) = -1/2.
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