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Cosine Graph: y = cos x

A clear walkthrough of the cosine graph y = cos x: how it comes from the unit circle, its domain, range, period, amplitude, and symmetry, plus how transformations shift, stretch, or compress it.

What is the cosine graph?

The cosine graph is the picture you get when you plot \(y = \cos x\) for every value of \(x\). It comes straight from the unit circle: as an angle \(x\) sweeps around the circle, \(\cos x\) is the horizontal coordinate of the point on the circle at that angle. If you're not yet comfortable with that idea, it helps to first review how the cosine ratio connects an angle to a side length in a right triangle, since the graph is really just that same ratio tracked continuously as the angle changes.

Because angles can be measured in degrees or radians, and the standard cosine graph uses radians, it's worth being confident with how to convert between degrees and radians before working through the examples below.

The graph of y = cos x

Here is the basic graph of \(y = \cos x\) over one full cycle in each direction. Notice it starts at its highest point when \(x = 0\), then falls, rises, and repeats.

Graph of \(y = \cos x\) from \(-2\pi\) to \(2\pi\)

A table of key values makes the shape easier to see:

\(x\)	0	\(\pi/2\)	\(\pi\)	\(3\pi/2\)	\(2\pi\)
\(\cos x\)	1	0	-1	0	1

Key features of the cosine graph

Domain: all real numbers, since you can find the cosine of any angle.
Range: \([-1, 1]\), because the graph never goes higher than 1 or lower than \(-1\).
Amplitude: 1, which is half the distance between the maximum and minimum values.
Period: \(2\pi\) radians (or 360 degrees), meaning the pattern repeats every \(2\pi\) units of \(x\).
Symmetry: cosine is an even function, so \(\cos(-x) = \cos x\) and the graph is a mirror image of itself across the y-axis.
Maximums and minimums: maximum value of 1 occurs at \(x = 0, \pm 2\pi, \pm 4\pi, \dots\); minimum value of \(-1\) occurs at \(x = \pm \pi, \pm 3\pi, \dots\).

The cosine graph looks almost identical to the sine graph, just shifted along the x-axis, and the two are often studied side by side. If you want to work backward from a picture of a wave to figure out whether it represents sine or cosine (and with what amplitude, period, or shift), that skill is covered in writing trig equations from graphs.

Transformations of y = cos x

Many problems ask you to graph a transformed cosine curve written as \(y = A\cos(Bx - C) + D\). Each letter changes the graph in a specific way:

\(A\) changes the amplitude to \(|A|\) (and flips the graph vertically if \(A\) is negative).
\(B\) changes the period to \(\frac{2\pi}{|B|}\); a larger \(B\) squeezes the graph horizontally.
\(C\) shifts the graph left or right (a phase shift of \(\frac{C}{B}\)).
\(D\) shifts the whole graph up or down.

Worked example

Graph \(y = 3\cos(2x)\) and describe how it differs from \(y = \cos x\).

Here \(A = 3\) and \(B = 2\), so the amplitude is \(|A| = 3\) and the period is \(\frac{2\pi}{2} = \pi\). That means the wave now reaches up to 3 and down to \(-3\), and it completes a full cycle in half the horizontal distance of the basic graph.

Graph of \(y = 3\cos(2x)\), showing a larger amplitude and shorter period than \(y = \cos x\)

The maximum value of 3 occurs at \(x = 0\), and because the period is \(\pi\) rather than \(2\pi\), the next maximum occurs at \(x = \pi\) instead of \(x = 2\pi\). Comparing this to the basic graph is a quick way to check that both the amplitude and period were identified correctly.

Cosine and its reciprocal graphs

Once the cosine graph feels familiar, it's worth seeing how its reciprocal function behaves: the sec x graph is built directly from \(y = \cos x\), since \(\sec x = \frac{1}{\cos x}\), producing curves and asymptotes wherever cosine equals zero.

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