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Trigonometry can feel abstract, but the unit circle turns those mysteries into concrete geometry. By placing a circle of radius 1 at the origin of a coordinate system, every trigonometric value becomes simply a point’s x‑ or y‑coordinate.
The unit circle has radius 1. Angles are measured from the point (1, 0) on the positive x‑axis and increase counter‑clockwise. For any angle θ:
A unit circle is simply a circle whose radius is exactly one unit. That one unit can be meters, feet, inches—any measurement; the key is that the radius is 1. Because of this, the circle’s circumference and area become simple multiples of π, and many trigonometric formulas reduce to pure numbers.
Place the circle so its center coincides with the origin of a Cartesian plane. The circle intersects the positive x‑axis at (1, 0). By convention, we start measuring angles from that point and move counter‑clockwise. Thus, the point (1, 0) corresponds to 0°, (0, 1) to 90°, (‑1, 0) to 180°, and (0, ‑1) to 270° (or –90°).
In elementary courses, sin, cos, and tan are introduced through right triangles:
\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
On the unit circle the hypotenuse is always 1, so the equations simplify to:
\(\sin\theta = \text{opposite}\)
\(\cos\theta = \text{adjacent}\)
If we draw a radius that makes an angle θ with the positive x‑axis, the “opposite” side is the y‑coordinate and the “adjacent” side is the x‑coordinate of the point where the radius meets the circle. Consequently, sin θ is the y‑coordinate and cos θ is the x‑coordinate. This explains why sin 0° = 0 and cos 0° = 1, or sin 90° = 1 and cos 90° = 0.
Negative angles are handled naturally: a clockwise rotation from the starting point shares the same x‑coordinate as the corresponding positive angle but flips the sign of the y‑coordinate. Hence:
\(\cos(-\theta) = \cos\theta\)
\(\sin(-\theta) = -\sin\theta\)
Using the circle definitions of sin and cos, tan simplifies to the ratio of the y‑coordinate to the x‑coordinate:
\(\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}\)
This form makes it clear why tan is undefined at 90° (or 270°), where x = 0, because division by zero is impossible.
When you view the unit circle, the x‑coordinate varies smoothly from 1 down to –1 as you move from 0° to 180°, then back up to 1 by 360°. The sine function follows the same pattern but reaches its peak of 1 at 90° first. Therefore, sin and cos are 90° out of phase. Tangent, being the ratio y/x, has vertical asymptotes where x = 0, producing the familiar repeating pattern with undefined points at odd multiples of 90°.
