Digit Math Mobile Website top of page banner image

Mobile Math Website

The Trigonometry of a Cosine Graph

The graph of a cosine function is periodic with period 2π and is similar to a graph of the sine function, with the difference that cosine function is shifted π/2 units left of the sine function for every corresponding value.

Period function of cosine and sine (red line) showing π/2 radian units shift:

The cosine function always grpahs π/2 radian units left of the sine function (red line).

To understand this π/2 radian units shift we equate five different degree measurements of cosine to sine:

cos 0° = 1 = sin 90° = sin (0 + 90)°
cos 30° = √3 / 2 = sin 60° =
sin 120° = sin (30 + 90)°
cos 45° = √2 / 2 = sin 45° =
sin 135° = sin (45 + 90)°
cos 60° = ½ = sin 30° =
sin 150° = sin (60 + 90)°
cos 90° = 0 = sin 180° =
sin (90 + 90)°

Observe that in each value of corresponding cosine values equated to sine values that:

cos A° = sin (A + 90)°
or, in radian measure,
cos θ = sin (θ + π / 2)

… and it is this relationship that allows us to draw the cosine graph by shifting the sine graph by π/2 units to the left.

The π/2 shift of sine that equates the sine graph to the cosine graph can also be shown on a unit circle.

This unit circle shows the cos θ = sin α =
sin (π − α) = sin (θ + π/2):

The π/2 shift of sine that equates a sine graph to a cosine graph is shown on this unit circle.

Top of Page About Us Privacy