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 pairing runs through all of trigonometry), and it sits among the wider family of trigonometric functions.

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.

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.

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.