The Unit Circle

To define the trigonometric functions for every real number, we begin with the unit circle — a circle centred at the origin with radius 1.

Given any real number $t$, we measure an arc of length $|t|$ along the circle starting from the point $(1,0)$: counter-clockwise when $t > 0$ and clockwise when $t < 0$. The terminal point of this arc is some point $P = (x, y)$ on the circle. We then define:

In words: cosine is the $x$-coordinate and sine is the $y$-coordinate of the point on the unit circle corresponding to the arc length (or angle) $t$.

Consistency with Right-Triangle Definitions

For an acute angle $\theta$ (between $0$ and $\frac{\pi}{2}$), drop a perpendicular from $P$ to the $x$-axis. This creates a right triangle whose:

• Hypotenuse = radius = 1
• Adjacent side = $x$-coordinate
• Opposite side = $y$-coordinate

So $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{1} = x$ and $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{1} = y$, which matches the unit-circle definition exactly.

Why the Circle Definition Extends to All Real Numbers

The right-triangle definition only works for angles between $0°$ and $90°$ (you can't have a triangle with a $200°$ angle). The unit circle removes this limitation: every real number $t$ determines a unique point on the circle, so $\sin t$ and $\cos t$ are defined for all $t \in \mathbb{R}$. Moreover, because the arc can wind around the circle multiple times, the functions naturally become periodic.

Interactive: drag the slider to move the point around the unit circle and see how $\cos t$ and $\sin t$ change.