The Unit Circle | Moosa Academy

Right triangles only give trigonometric values for acute angles. The unit circle lifts that limit: every angle, of any size and either direction, lands on a point of the circle, and that point is the pair of values $(\cos\theta,\ \sin\theta)$ .

Concept Definition and coordinates

The unit circle is centred at the origin with radius 1, so its equation is $x^2 + y^2 = 1$ .

cos θ is the horizontal coordinate of the point.

sin θ is the vertical coordinate.

A positive angle turns counter-clockwise; a negative angle turns clockwise.

Because the radius is exactly one unit, no division is needed: the ratio $\dfrac{\text{opp}}{\text{hyp}}$ becomes simply the height $y$ , and $\dfrac{\text{adj}}{\text{hyp}}$ becomes the width $x$ .

Reference Signs in the four quadrants

Quadrant	Range	cos θ (x)	sin θ (y)
First	0° – 90°	positive	positive
Second	90° – 180°	negative	positive
Third	180° – 270°	negative	negative
Fourth	270° – 360°	positive	negative

Example Locating a point at 70°

A body moves along the unit circle and stops at an angle of 70°. Find its coordinates and name the quadrant.

x = \cos 70° \approx 0.342

y = \sin 70° \approx 0.940

70° lies between 0° and 90°, so the point is in the first quadrant — and indeed both coordinates are positive.

⟹ the point is (0.342, 0.940), first quadrant

Summary

The unit circle is centred at the origin with radius 1: x² + y² = 1.
The point at angle θ is (cos θ, sin θ) — the definitions now work for any angle.
cos θ carries the sign of the x-coordinate; sin θ carries the sign of the y-coordinate.
Positive angles turn counter-clockwise, negative angles clockwise.