# Cosine Function - Graph, Properties & Examples

TL;DR

The cosine function gives the xxx-coordinate of a point on the unit circle, or the ratio \( \frac{\text{adjacent}}{\text{hypotenuse}} \) in a right triangle. This article covers the definition, the cosine graph (period \(2\pi\), range \([-1,1]\)), why cosine is an even function, its quadrant signs, key values, and worked examples.

## What Is the Cosine Function?

The **cosine function**, written \( \cos\theta \), is one of the three primary trigonometric functions. It has two equivalent definitions, and a student needs both.

- **Right-triangle definition.** For an acute angle \( \theta \) in a right triangle, \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \). This is the SOH-CAH-TOA "CAH" — Cosine, Adjacent, Hypotenuse.

- **Unit-circle definition.** For any angle \( \theta \), place the angle at the centre of a circle of radius 1. The point where the terminal ray meets the circle has coordinates \( (\cos\theta,\sin\theta) \). So **cosine is the xxx-coordinate** of that point.

The triangle definition handles angles between 0° and 90°. The unit-circle definition extends cosine to every angle, including obtuse, reflex, and negative ones — which is what lets cosine become a wave rather than just a ratio. Cosine is the complement of sine (the [sine and cosine](/content/math/formula/sin-cos/index.html) pairing runs through all of trigonometry), and it sits among the wider family of [trigonometric functions](/content/math/trigonometry/trigonometric-functions/index.html).

### Computing the Same Value Two Ways

Take \( \theta=60° \). From the right triangle (a 30-60-90 with hypotenuse 2, adjacent side 1):

\[ \cos 60° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2} \]

From the unit circle, the point at 60° is \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \), so the xxx-coordinate is \( \frac{1}{2} \). Same answer, two anchors. Holding both prevents the common gap where cosine feels like "a triangle ratio" in one chapter and "a wave" in the next, never the same object.

## What Are the Properties of the Cosine Function?

The behaviour of \( \cos\theta \) is fixed by a handful of properties, each readable straight off the graph above.

- **Domain:** all real numbers — you can feed cosine any angle.
- **Range:** \([-1,1]\) — the output never escapes this band.
- **Period:** \( 2\pi \) (or 360°). The wave repeats every full turn: \( \cos(\theta + 2\pi) = \cos\theta \).
- **Amplitude:** 1 — the distance from the centre line to a peak.
- **Even function:** \( \cos(-\theta) = \cos\theta \). The graph is a mirror image across the y-axis.
- **Maximum / minimum:** +1 at \( \theta=0,2\pi,… \) and −1 at \( \theta=\pi,3\pi,… \)

### Cosine Signs by Quadrant

Because cosine is the xxx-coordinate, its sign follows the sign of xxx around the circle.

| Quadrant | Angle range | Sign of \( \cos\theta \) |
| --- | --- | --- |
| I | 0° to 90° | Positive |
| II | 90° to 180° | Negative |
| III | 180° to 270° | Negative |
| IV | 270° to 360° | Positive |

The word "quadrant" just names one of the four regions the axes cut the plane into; cosine is positive wherever a point sits to the right of the y-axis.

## Key Cosine Values

These special-angle values are worth knowing cold; they recur throughout trigonometry and feed the [trigonometric table](/content/math/trigonometry/trigonometric-table/index.html).

| \( \theta \) | 0° | 30° | 45° | 60° | 90° | 180° |
| --- | --- | --- | --- | --- | --- | --- |
| \( \cos\theta \) | 1 | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{2} \) | 0 | −1 |

## Examples of the Cosine Function

### Example 1

**A right triangle has an adjacent side of 4 and a hypotenuse of 5. Find \( \cos\theta \).**

\[ \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]

**Final answer:** \( \cos\theta = \frac{4}{5} \).

### Example 2

**Evaluate \( \cos(-60°) \). First instinct, then the correct route.**

Check it against the graph. Cosine is symmetric about the y-axis — the curve at -60° sits at the same height as at +60°, both above the axis. A negative answer would put it below. So the instinct is wrong.

The rescue is the even-function property: \( \cos(-\theta) = \cos\theta \).

\[ \cos(-60°) = \cos 60° = \frac{1}{2} \]

**Final answer:** \( \cos(-60°) = \frac{1}{2} \).

### Example 3

**Find \( \cos 120° \) using a reference angle.**

\( 120° \) lands in Quadrant II, where cosine is negative. Its reference angle is \( 180°−120°=60° \).

\[ \cos 120° = -\cos 60° = -\frac{1}{2} \]

**Final answer:** \( \cos 120° = -\frac{1}{2} \).

### Example 4

**The amplitude of \( y=3\cos x \) is what, and what is its range?**

Amplitude = 3, so the range is \([-3,3]\).

**Final answer:** amplitude 3, range \([-3,3]\).

### Example 5

**Find the period of \( y=\cos(2x) \).**

The period of \( \cos(bx) \) is \( \frac{2\pi}{b} \). Here \( b=2 \):

\[ \text{Period} = \frac{2\pi}{2} = \pi \]

**Final answer:** period = \( \pi \).

### Example 6

**Verify the Pythagorean identity at \( \theta=45° \).**

The identity \( \cos^2\theta + \sin^2\theta = 1 \) holds for every angle. At 45°, \( \cos 45° = \sin 45° = \frac{1}{\sqrt{2}} \):

\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 1 \checkmark \]

**Final answer:** the identity holds.

## Why the Cosine Function Matters - "Sideways Position Under Rotation"

Cosine exists to describe one specific thing well: how far a rotating object has travelled in the horizontal direction. That single job is why the function is everywhere periodic motion is.

- **Signal processing.** Any sound or radio wave can be built from cosine waves of different frequencies — the basis of the Fourier transform behind audio and image compression.
- **Physics of waves and oscillation.** A swinging pendulum or a mass on a spring has a position that follows \( x(t)=A\cos(\omega t) \).
- **The dot product and angles.** In vectors, \( \cos\theta \) measures how aligned two directions are.

## Common Mistakes With the Cosine Function

### Mistake 1: Treating \( \cos(-\theta) \) as \(-\cos\theta \)

**Correct way:**  \( \cos(-\theta) = +\cos\theta \).

### Mistake 2: Forgetting the quadrant sign

**Correct way:** Account for the quadrant when using the reference angle.

### Mistake 3: Confusing amplitude with period

**Correct way:** The outside coefficient sets amplitude; the inside coefficient sets the period \( \frac{2\pi}{b} \).

## Key Takeaways

- The **cosine function** is \( \frac{\text{adjacent}}{\text{hypotenuse}} \) in a triangle and the xxx-coordinate on the unit circle.
- Its graph has period \(2\pi\), amplitude 1, and range \([-1,1]\).
- Cosine is **even**: \( \cos(-\theta) = \cos\theta \).
- It is positive in Quadrants I and IV, negative in II and III.