AI Tutor
If a satellite travels along a circular orbit of radius 1, where is it located at any second?
Instead of recalculating right triangles from scratch every time, a single coordinate pair tells you both its exact horizontal and vertical position.

The Unit Circle Coordinates

A unit circle is a circle centered at (0,0)(0,0) with a radius of exactly r=1r = 1.
When an angle ΞΈ\theta is in standard position, its terminal ray intersects the circle at a single point (x,y)(x, y).
Because cos⁑θ=adjacenthypotenuse=x1=x\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{1} = x and sin⁑θ=oppositehypotenuse=y1=y\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{1} = y, every point on the unit circle is simply (cos⁑θ,sin⁑θ)(\cos\theta, \sin\theta).
Swiss mathematician Leonhard Euler popularized this coordinate approach in 1748, turning trigonometry from a study of triangle side lengths into a study of circular functions.
What happens when the angle lands directly on one of the coordinate axes?

Quadrantal Angles

A quadrantal angle is an angle whose terminal side lies along the xx-axis or yy-axis, such as 00, Ο€2\frac{\pi}{2}, Ο€\pi, or 3Ο€2\frac{3\pi}{2} radians.
To find their trigonometric values, read the (x,y)(x, y) coordinates where each axis meets the unit circle.