AI Tutor
If triangle trigonometry only works for acute angles inside a right triangle, how can you find cos⁑(150∘)\cos(150^\circ) or sin⁑(4Ο€3)\sin\left(\frac{4\pi}{3}\right)?
The unit circle solves this by turning trigonometry from triangle side ratios into coordinates on a circle of radius 11 centered at the origin (0,0)(0, 0).
In 150 BCE, Greek astronomer Hipparchus created chord tables to map planetary orbits, laying the groundwork for mapping angles directly to circular coordinates.
How do standard xx and yy coordinates relate to our familiar sine and cosine functions?

The (x,y)(x, y) Connection

When you draw an angle ΞΈ\theta in standard position, its terminal ray intersects the unit circle at a point (x,y)(x, y).
Because the hypotenuse is r=1r = 1, basic right-triangle math gives cos⁑θ=x1=x\cos\theta = \frac{x}{1} = x and sin⁑θ=y1=y\sin\theta = \frac{y}{1} = y.
Once you know that every point is (cos⁑θ,sin⁑θ)(\cos\theta, \sin\theta), you can define all six trigonometric functions directly from xx and yy.

The Six Trig Definitions

The primary functions are cos⁑θ=x\cos\theta = x, sin⁑θ=y\sin\theta = y, and their ratio tan⁑θ=yx\tan\theta = \frac{y}{x} (where xβ‰ 0x \neq 0).
The three reciprocal functions flip these values: sec⁑θ=1x\sec\theta = \frac{1}{x}, csc⁑θ=1y\csc\theta = \frac{1}{y}, and cot⁑θ=xy\cot\theta = \frac{x}{y}.