02 // Visual Math & Finance Solvers(x, y) = (R·cos θ, R·sin θ), cos²θ + sin²θ = 1
Interactive Unit Circle & Six Trig Functions Explorer
The unit circle defines a circle of radius $R = 1$ centered at the Cartesian origin $(0,0)$, where any central angle $\theta$ intersects the circumference at $(x, y) = (\cos\theta, \sin\theta)$ and obeys $\cos^2\theta + \sin^2\theta = 1$.
(x, y) = (R·cos θ, R·sin θ), cos²θ + sin²θ = 1Inputs & Parameters (3)
°
0°±1360°
u
0.5u±0.255u
×
1×±14×
R·cos(θ) [X-Base]
0.7071
cos(45°) = 0.7071
R·sin(θ) [Y-Altitude]
0.7071
sin(45°) = 0.7071
tan(θ) = sin / cos
1
cot(θ) = 1
Radians & Arc Length
0.7854 rad
0.25π rad | Arc s = 0.785
20-Row Comparison Table
Compare across:
| # | Central Angle (θ) (°) | y = R·sin(θ) | R·cos(θ) | tan(θ) |
|---|---|---|---|---|
| 01 | 0 ° | 0 | 1 | 0 |
| 02 | 19 ° | 0.326 | 0.9455 | 0.3443 |
| 03 | 38 ° | 0.616 | 0.788 | 0.7813 |
| 04 | 57 ° | 0.839 | 0.5446 | 1.5399 |
| 05 | 76 ° | 0.97 | 0.2419 | 4.0108 |
| 06 | 95 ° | 0.996 | -0.0872 | -11.4301 |
| 07 | 114 ° | 0.914 | -0.4067 | -2.246 |
| 08 | 133 ° | 0.731 | -0.682 | -1.0724 |
| 09 | 152 ° | 0.469 | -0.8829 | -0.5317 |
| 10 | 171 ° | 0.156 | -0.9877 | -0.1584 |
| 11 | 189 ° | -0.156 | -0.9877 | 0.1584 |
| 12 | 208 ° | -0.469 | -0.8829 | 0.5317 |
| 13 | 227 ° | -0.731 | -0.682 | 1.0724 |
| 14 | 246 ° | -0.914 | -0.4067 | 2.246 |
| 15 | 265 ° | -0.996 | -0.0872 | 11.4301 |
| 16 | 284 ° | -0.97 | 0.2419 | -4.0108 |
| 17 | 303 ° | -0.839 | 0.5446 | -1.5399 |
| 18 | 322 ° | -0.616 | 0.788 | -0.7813 |
| 19 | 341 ° | -0.326 | 0.9455 | -0.3443 |
| 20 | 360 ° | -0 | 1 | -0 |
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Key Analytical Takeaways
- 01The horizontal base projection (Cobalt) equals $R\cos\theta$, while the vertical altitude drop (Mineral Teal) equals $R\sin\theta$.
- 02At special benchmark angles ($30^\circ, 45^\circ, 60^\circ$), exact radical ratios ($1/2, \sqrt{2}/2, \sqrt{3}/2$) appear automatically.
- 03Quadrant sign mnemonic 'All Students Take Calculus' tracks which functions remain positive across Quadrants I, II, III, and IV.
Frequently Asked Questions
Why is the radius set to 1 on the standard unit circle?+
When the hypotenuse R = 1, dividing opposite or adjacent sides by the hypotenuse simplifies directly to y = sin(θ) and x = cos(θ) without extra scaling.
Why does tan(θ) become undefined at 90° and 270°?+
Because tan(θ) = sin(θ) / cos(θ), and at 90° and 270° the x-coordinate cos(θ) equals zero, making the tangent line parallel to the y-axis.
Which trig functions are positive in each quadrant (ASTC rule)?+
Following All-Students-Take-Calculus counterclockwise: all six are positive in Q1 (0°–90°), Sine/Cosecant in Q2 (90°–180°), Tangent/Cotangent in Q3 (180°–270°), and Cosine/Secant in Q4 (270°–360°).