Cosine (cos)

What is cosine

The cosine of an angle is the ratio of the adjacent leg to the hypotenuse. On the unit circle, cosine is the horizontal (x) coordinate of the point.

Domain and properties

The domain is all real numbers, the range is [−1, 1]. The function is periodic with period 360° and even: cos(−α) = cos(α). Cosine is just a sine shifted by 90°.

Notable values

cos 0° = 1, cos 30° = √3/2 ≈ 0.866, cos 45° = √2/2 ≈ 0.707, cos 60° = 0.5, cos 90° = 0

Where cosine is used in practice

• Combining forces and vectors – the projection of a force onto a direction is computed with cosine.
• The law of cosines for finding a side of a triangle that is not right-angled.
• GPS and navigation – distances on a sphere are computed via cosines of geographic angles.

The unit circle and graphs

The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.

Relationships between the functions

tan α = sin α / cos α  •  cot α = 1 / tan α  •  sin²α + cos²α = 1