Cosine Explained: The Horizontal Partner to Sine

Cosine is horizontal position.

When a point rotates around a circle, cosine tells you how far right it is. At 3 o'clock (angle 0°), cosine equals 1. At 9 o'clock (angle 180°), cosine equals -1. At the top and bottom (90° and 270°), cosine equals 0.

Here's the unlock: cosine isn't a different function from sine. It's sine shifted by 90°. When sine is at its maximum, cosine is at zero. When sine is at zero, cosine is at maximum. They're the same wave, just out of phase.

Cosine is what sine looks like a quarter turn later.

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.

cos θ = the x-coordinate of that point

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

Cosine and Sine Together

The point at angle θ on the unit circle has coordinates:

(cos θ, sin θ)

Cosine gives the x-coordinate. Sine gives the y-coordinate. Together they completely specify the position.

This is why cos²θ + sin²θ = 1. The coordinates of any point on the unit circle satisfy x² + y² = 1.

The Cosine Wave

Plot cosine values as the angle increases:

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

Compare to sine: sine starts at 0 and rises, cosine starts at 1 and falls. Otherwise identical shapes.

Cosine leads sine by 90°: cos θ = sin(θ + 90°)

Key Values to Know

cos 0° = 1 (point on positive x-axis)

cos 30° = √3/2 ≈ 0.866 (almost all the way right)

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

cos 60° = 1/2 (halfway right)

cos 90° = 0 (point on y-axis)

Notice: cos 30° = sin 60°, cos 45° = sin 45°, cos 60° = sin 30°.

Complementary angles: cos θ = sin(90° - θ)

Why "Adjacent Over Hypotenuse" Works

In a right triangle with angle θ:

cos θ = adjacent side / hypotenuse

Why? Place the triangle inside a unit circle. The hypotenuse becomes the radius (1), and the adjacent side becomes the x-coordinate.

adjacent / hypotenuse = x / 1 = x = cos θ

Cosine Is Even: cos(-θ) = cos(θ)

Rotating clockwise (negative angle) puts you at the same x-coordinate. The y-coordinate flips, but x stays the same.

cos(-30°) = cos(30°) = √3/2

This makes cosine an even function — symmetric about the y-axis.

Contrast with sine, which is odd: sin(-θ) = -sin(θ).

Cosine of Angles Greater Than 90°

Second quadrant (90° to 180°): x is negative, so cosine is negative. cos 120° = -cos 60° = -0.5

Third quadrant (180° to 270°): x is still negative. cos 210° = -cos 30° = -√3/2

Fourth quadrant (270° to 360°): x is positive again. cos 330° = cos 30° = √3/2

Derivatives and Calculus

In calculus:

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

The rate of change of cosine is negative sine. This is because when you're moving counterclockwise at the rightmost point (cosine at max), your vertical velocity is zero and about to become positive (negative of the derivative).

The derivatives cycle:

• d/dx sin(x) = cos(x)
• d/dx cos(x) = -sin(x)
• d/dx (-sin(x)) = -cos(x)
• d/dx (-cos(x)) = sin(x)

The Inverse: Arccosine

arccos (or cos⁻¹) reverses cosine. Given an x-value, it returns the angle.

arccos(0.5) = 60° because cos(60°) = 0.5

But multiple angles have the same cosine: cos(60°) = cos(300°) = 0.5

By convention, arccos returns values between 0° and 180° (or 0 to π radians).

Why Cosine Appears with Sine

Physics problems rarely use just sine or just cosine. They use both, because real rotation happens in two dimensions.

Circular motion: x(t) = r cos(ωt), y(t) = r sin(ωt)

Waves: often written as A cos(ωt + φ) or A sin(ωt + φ)

Any phase-shifted sine can be written as a cosine, and vice versa. They're the same wave, differently timed.

Euler's Connection

Euler's formula links cosine and sine to complex exponentials:

e^(iθ) = cos θ + i sin θ

This means: cos θ = (e^(iθ) + e^(-iθ)) / 2 sin θ = (e^(iθ) - e^(-iθ)) / 2i

Cosine is the "real part" of rotation in the complex plane. Sine is the "imaginary part."

The Core Insight

Cosine is the horizontal coordinate of rotation — sine's partner, shifted 90°.

Where sine measures how high the rotating point is, cosine measures how far right. Together they fully describe position on the circle.

Every trig identity, every wave equation, every oscillation formula ultimately comes from this pair of coordinates. Sine and cosine are not two separate things to memorize — they're two views of the same rotation.

Master one, and you've mastered both.

Part 3 of the Trigonometry series.

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