The Unit Circle: Why Radius 1 Changes Everything

The unit circle is a circle with radius 1 centered at the origin.

That's it. Nothing fancy. But this simple choice — radius exactly 1 — transforms trigonometry from messy ratios into clean coordinates.

Here's the unlock: on the unit circle, sine and cosine ARE the coordinates. The point at angle θ sits at (cos θ, sin θ). No division needed. No hypotenuse to factor out. The coordinates just ARE the trig values.

This is why the unit circle is the central object of trigonometry. Everything else is scaling.

Why Radius 1 Matters

On a circle with radius r, the point at angle θ has coordinates: (r cos θ, r sin θ)

On the unit circle (r = 1): (cos θ, sin θ)

The radius drops out. Cosine is literally the x-coordinate. Sine is literally the y-coordinate.

Every trig calculation becomes coordinate reading.

The Setup

Draw the coordinate axes. Draw a circle of radius 1 centered at (0, 0).

The circle passes through four special points:

• (1, 0) at angle 0° (3 o'clock)
• (0, 1) at angle 90° (12 o'clock)
• (-1, 0) at angle 180° (9 o'clock)
• (0, -1) at angle 270° (6 o'clock)

Angles are measured counterclockwise from the positive x-axis.

Reading Trig Values Directly

Point at 0°: coordinates (1, 0) Therefore: cos 0° = 1, sin 0° = 0

Point at 90°: coordinates (0, 1) Therefore: cos 90° = 0, sin 90° = 1

Point at 180°: coordinates (-1, 0) Therefore: cos 180° = -1, sin 180° = 0

Point at 270°: coordinates (0, -1) Therefore: cos 270° = 0, sin 270° = -1

No formulas needed. Just read the coordinates.

The Special Angles

Beyond the axis points, certain angles have coordinates you should know:

30° (π/6): (√3/2, 1/2) cos 30° = √3/2 ≈ 0.866 sin 30° = 1/2 = 0.5

45° (π/4): (√2/2, √2/2) cos 45° = sin 45° = √2/2 ≈ 0.707

60° (π/3): (1/2, √3/2) cos 60° = 1/2 = 0.5 sin 60° = √3/2 ≈ 0.866

Notice: the coordinates of 30° and 60° are swapped. This is because 30° and 60° are complementary (they sum to 90°).

The Pattern: 0, 1/2, √2/2, √3/2, 1

Look at sine values from 0° to 90°:

• sin 0° = 0
• sin 30° = 1/2
• sin 45° = √2/2
• sin 60° = √3/2
• sin 90° = 1

Memory trick: √0/2, √1/2, √2/2, √3/2, √4/2

Cosine follows the same pattern in reverse (from 1 down to 0).

The Four Quadrants

The coordinate axes divide the plane into four quadrants:

Quadrant I (0° to 90°): x > 0, y > 0 Both cosine and sine are positive.

Quadrant II (90° to 180°): x < 0, y > 0 Cosine negative, sine positive.

Quadrant III (180° to 270°): x < 0, y < 0 Both cosine and sine are negative.

Quadrant IV (270° to 360°): x > 0, y < 0 Cosine positive, sine negative.

Mnemonic: "All Students Take Calculus" — All positive in I, Sine in II, Tangent in III, Cosine in IV.

Reference Angles

Every angle has a reference angle — the acute angle formed with the x-axis.

For 150°: reference angle is 180° - 150° = 30° For 210°: reference angle is 210° - 180° = 30° For 330°: reference angle is 360° - 330° = 30°

The reference angle determines the magnitude of the trig values. The quadrant determines the signs.

sin 150° = sin 30° = 0.5 (Quadrant II, sine positive) sin 210° = -sin 30° = -0.5 (Quadrant III, sine negative) sin 330° = -sin 30° = -0.5 (Quadrant IV, sine negative)

The Pythagorean Identity

Every point on the unit circle satisfies x² + y² = 1.

Since x = cos θ and y = sin θ:

cos²θ + sin²θ = 1

This isn't something to memorize as a separate fact. It's the equation of the unit circle written in trig notation.

Tangent on the Unit Circle

tan θ = sin θ / cos θ = y/x

This is the slope of the line from origin to the point.

Where x = 0 (at 90° and 270°), tangent is undefined — vertical lines have no slope.

Beyond 360°

Angles larger than 360° just wrap around:

• 390° = 360° + 30° → same as 30°
• 720° = 2 × 360° → same as 0°

Negative angles go clockwise:

• -30° = 330° (same position, opposite direction)
• -90° = 270°

The unit circle handles all angles, positive, negative, or huge.

Radians Make the Circle Natural

The unit circle is why radians are natural.

One radian is the angle where the arc length equals the radius. On a unit circle (radius 1), a radian angle sweeps out an arc of length 1.

Full circle: circumference = 2π(1) = 2π So a full rotation is 2π radians.

Arc length = radius × angle (in radians) On unit circle: arc length = angle

Why the Unit Circle Is Central

Every trig problem can be solved on the unit circle:

1. Find the angle on the circle
2. Read off the coordinates
3. Scale if your radius isn't 1

For a circle of radius 5: x = 5 cos θ, y = 5 sin θ

The unit circle gives you the ratios. Multiply by your radius.

The Core Insight

The unit circle eliminates the clutter.

When radius = 1, coordinates ARE trig values. No ratios to compute, no hypotenuses to divide. Sine is height. Cosine is horizontal position. Period.

This is why every trig course should start with the unit circle, not SOH CAH TOA. The ratios are just what happens when you have a different radius and need to scale back to 1.

Master the unit circle — visualize where the point is, read its coordinates — and trigonometry stops being mysterious.

Part 5 of the Trigonometry series.

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