Graph of cosine: y = cos x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The cosine graph is the sine wave shifted a quarter period to the left. It starts at its highest point, (0, 1), falls to −1 at x = π and is back at 1 by x = 2π, repeating forever in both directions.

Open in calculator

The graph of

y=cos(x)

for −7 ≤ x ≤ 7. Dots mark its roots, turning points and y-intercept.

Key features

Domain: All real numbers
Range: −1 ≤ y ≤ 1
Period: 2π ≈ 6.2832
Amplitude: 1
Zeros: x = π/2 + nπ
Maximum: y = 1 at x = 2nπ
Minimum: y = −1 at x = π + 2nπ
Symmetry: Even: cos(−x) = cos x, so the graph is symmetric about the y-axis
y-intercept: None
Derivative: −sin(x)
Antiderivative: sin(x)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of cos x looks like

From its peak at (0, 1), the graph falls through zero at x = π/2, bottoms out at (π, −1), climbs back through zero at x = 3π/2 and returns to the peak at (2π, 1). That is one period.

Cosine is an even function, so the left half of the graph is a mirror image of the right half across the y-axis. Its slope is −sin x: zero at the peaks and troughs, and steepest, at ±1, where the curve crosses the axis.

Each value of cos x is the horizontal coordinate of a point moving round the unit circle through an angle x, and sin x is its vertical coordinate. That is why cos²x + sin²x = 1 for every x, and why the two graphs are copies of each other shifted by π/2.

Points of interest

Found numerically to about twelve digits in the window −7 ≤ x ≤ 7, then shown in exact form where one exists.

x-intercepts (roots): x = −4.7124, x = −1.5708, x = 1.5708, x = 4.7124
Local maxima: (−6.2832, 1), (0, 1), (6.2832, 1)
Local minima: (−3.1416, −1), (3.1416, −1)
Inflection points: (−3π/2, 0), (−π/2, 0), (π/2, 0), (3π/2, 0)

Table of values

A few points to plot, with exact values where they are simple.

Values of cos x
x	y = cos x
0	1
π/6	√3/2
π/4	√2/2
π/3	1/2
π/2	0
2π/3	−1/2
π	−1
3π/2	0
2π	1

How to graph cos x by hand

1
Mark the x-axis in steps of π/2.
2
Plot (0, 1), (π/2, 0), (π, −1), (3π/2, 0) and (2π, 1).
3
Join them with a smooth wave, flat at the top and bottom and slanted where it crosses the axis.
4
Mirror the shape to the left of the y-axis, since cos(−x) = cos x, and repeat it every 2π.

Transformations

How the graph moves when you change the formula. The original is cos x in blue.

Open in calculator

y = 3cos(x) Amplitude 3: the range becomes −3 ≤ y ≤ 3.
y = cos(x/2) Period 4π: the wave is stretched horizontally to twice its width.
y = -cos(x) + 1 Reflected in the x-axis, then shifted up by 1, so it runs between 0 and 2.

Common mistakes

Starting the cosine graph at the origin. cos 0 = 1, so the graph crosses the y-axis at (0, 1).
Forgetting that cos(bx) has period 2π/b: cos 2x repeats every π.

Where it shows up

The horizontal position of a point on a rotating wheel, crank or Ferris wheel.
Projections: the component of a vector v along a direction at angle θ is |v| cos θ, the basis of the dot product.
Damped oscillations such as a car suspension settling, modelled by e^(−kt) cos(ωt).