Interactive Unit Circle
Visualize Sine, Cosine, and Tangent. Drag the point to see how trigonometric functions relate to the circle.
Understand trigonometry intuitively with our **Interactive Unit Circle**. See exactly why $\sin(90^\circ) = 1$ and $\cos(90^\circ) = 0$.
Control Panel
Sine & Cosine Waves
Trig Identity Checklist
What is Interactive Unit Circle?
What is the Unit Circle?
It is a circle with a radius of exactly **1**, centered at the origin (0,0) of the Cartesian coordinate system. It allows us to define trigonometric functions for any angle $\theta$, not just for right-angled triangles.
Any point $(x, y)$ on the circle corresponds to $(\cos \theta, \sin \theta)$.
Formula & Calculation
The coordinates of a point on the Unit Circle are defined by:
x-coordinate
x = cos(θ)y-coordinate
y = sin(θ)Tangent
tan(θ) = y / xPythagorean Identity
Since $x^2 + y^2 = r^2$ equals the radius ($r=1$):
sin²(θ) + cos²(θ) = 1
Example Calculation
Common Angles
Memorize these key points:
| Angle | Radians | Sin | Cos |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | π/6 | 1/2 | √3/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | √3/2 | 1/2 |
| 90° | π/2 | 1 | 0 |
Frequently Asked Questions
What happens to Tangent at 90°?
At 90 degrees, $x = \cos(90) = 0$. Since $\tan(\theta) = y/x$, we divide by zero, making Tangent undefined (infinity).
Why do we use Radians?
Radians relate the angle directly to the arc length. In calculus (derivatives/integrals of trig functions), using degrees introduces messy constants, whereas radians keep equations clean.