Sine Explained: The Height of a Point on a Spinning Circle

Sine is vertical position.

That's the whole thing. When a point rotates around a circle, sine tells you how high it is. At the top of the circle, sine equals 1. At the bottom, sine equals -1. In the middle (at 3 o'clock or 9 o'clock), sine equals 0.

Here's why this matters: anything that oscillates — bouncing, swinging, vibrating — traces out a sine wave. Because up-and-down motion is what you see when you watch circular motion from the side.

Sine isn't a formula. It's the shadow of a wheel.

On the Unit Circle

Place a point on a circle of radius 1 centered at the origin. Rotate it counterclockwise from the positive x-axis by angle θ.

sin θ = the y-coordinate of that point

That's the definition. Everything else follows.

At θ = 0° (starting position, 3 o'clock): sin 0° = 0 At θ = 90° (top, 12 o'clock): sin 90° = 1 At θ = 180° (left, 9 o'clock): sin 180° = 0 At θ = 270° (bottom, 6 o'clock): sin 270° = -1 At θ = 360° (back to start): sin 360° = 0

The Sine Wave

If you plot sine values as the angle increases, you get the famous wave shape.

Starting at 0, rising to 1, falling through 0 to -1, rising back to 0. One complete cycle every 360° (or 2π radians).

This wave appears everywhere because it describes the simplest form of oscillation — what happens when something rotates smoothly.

Key Values to Know

These values come from the geometry of the unit circle:

sin 0° = 0 (point on x-axis, no height)

sin 30° = 1/2 (halfway up)

sin 45° = √2/2 ≈ 0.707 (equal x and y coordinates)

sin 60° = √3/2 ≈ 0.866 (almost to top)

sin 90° = 1 (top of circle)

In radians:

• sin(π/6) = 1/2
• sin(π/4) = √2/2
• sin(π/3) = √3/2
• sin(π/2) = 1

Why "Opposite Over Hypotenuse" Works

In a right triangle with angle θ:

sin θ = opposite side / hypotenuse

Why? Because if you place that triangle inside a unit circle, the hypotenuse becomes the radius (1), and the opposite side becomes the y-coordinate.

opposite / hypotenuse = y / 1 = y = sin θ

The ratio definition is a consequence, not the foundation.

Sine Is Odd: sin(-θ) = -sin(θ)

Rotating clockwise (negative angle) puts you at the same x-coordinate but opposite y-coordinate.

sin(-30°) = -sin(30°) = -0.5

This makes sine an odd function — symmetric about the origin, not the y-axis.

Sine of Angles Greater Than 90°

The unit circle definition extends sine beyond right triangles.

Second quadrant (90° to 180°): y is still positive, so sine is positive. sin 120° = sin 60° = √3/2

Third quadrant (180° to 270°): y is negative, so sine is negative. sin 210° = -sin 30° = -0.5

Fourth quadrant (270° to 360°): y is still negative. sin 330° = -sin 30° = -0.5

The triangle definition only works for acute angles. The circle definition works for all angles.

Period and Amplitude

Period: Sine completes one full cycle every 360° (2π radians). After that, values repeat.

sin(θ + 360°) = sin θ

Amplitude: Sine ranges from -1 to 1. The amplitude is 1 (half the range from peak to trough).

For y = A sin(θ), the amplitude is A. For y = sin(Bθ), the period is 360°/B.

Derivatives and Calculus

In calculus:

d/dx sin(x) = cos(x) (when x is in radians)

The rate of change of sine is cosine. This is because the horizontal velocity of a rotating point is proportional to its horizontal position.

This clean relationship only works in radians — it's why mathematicians prefer them.

Sine in Right Triangles

For a right triangle with angle θ:

sin θ = (length of opposite side) / (length of hypotenuse)

If you know the angle and the hypotenuse, you can find the opposite side: opposite = hypotenuse × sin θ

If you know the opposite and hypotenuse, you can find the angle: θ = sin⁻¹(opposite / hypotenuse) = arcsin(opposite / hypotenuse)

The Inverse: Arcsine

arcsin (or sin⁻¹) reverses sine. Given a y-value, it returns the angle.

arcsin(0.5) = 30° because sin(30°) = 0.5

But there's a catch: many angles have the same sine value. sin(30°) = sin(150°) = 0.5

By convention, arcsin returns values between -90° and 90° (or -π/2 to π/2).

Why Sine Waves Describe Sound

Sound is pressure variation in air. A pure tone — a single frequency — is a sine wave of pressure.

When something vibrates at a steady rate, it pushes air in a sinusoidal pattern. Your eardrum responds to these pressure waves.

More complex sounds are combinations of sine waves at different frequencies. This is Fourier's great insight: any periodic wave can be decomposed into sines.

The Core Insight

Sine is the vertical coordinate of rotation.

On a unit circle, sin θ is simply how high the rotating point is. This makes sine the natural function for describing anything that oscillates vertically — which turns out to be most things that oscillate at all.

Pendulums swing through sine. Sound waves oscillate through sine. Light waves propagate through sine. Quantum mechanical wavefunctions evolve through sine.

The sine function isn't arbitrary. It's what smooth rotation looks like when viewed from the side.

Part 2 of the Trigonometry series.

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