Cosine (cos) Calculator

Calculate the cosine of any angle in degrees or radians with unit circle visualization and related trig values.

Enter Angle

Result

cos(60°)
0.5
60 degrees
Angle (degrees) 60
Angle (radians) 1.047198
cos(theta) 0.5
sin(theta) 0.866025
tan(theta) 1.732051
sec(theta) 2
Quadrant I

Step-by-Step Solution

cos(60 deg) = 0.5

Understanding the Cosine Function

The cosine function is one of the fundamental trigonometric functions. In a right triangle, cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. On the unit circle, cos(theta) gives the x-coordinate of the point at angle theta from the positive x-axis.

The cosine function is periodic with period 2pi (360 degrees) and has a range of [-1, 1]. It is an even function, meaning cos(-theta) = cos(theta).

Common Cosine Values

cos(0°) = 1

At 0 degrees (0 radians), cosine equals 1.

cos(0) = 1

cos(30°) = √3/2

At 30 degrees (pi/6 radians).

cos(pi/6) = 0.866025

cos(45°) = √2/2

At 45 degrees (pi/4 radians).

cos(pi/4) = 0.707107

cos(60°) = 1/2

At 60 degrees (pi/3 radians).

cos(pi/3) = 0.5

cos(90°) = 0

At 90 degrees (pi/2 radians), cosine equals 0.

cos(pi/2) = 0

cos(180°) = -1

At 180 degrees (pi radians), cosine equals -1.

cos(pi) = -1

Cosine Identities

The cosine function satisfies many important identities:

  • Pythagorean identity: cos^2(theta) + sin^2(theta) = 1
  • Even function: cos(-theta) = cos(theta)
  • Complementary angle: cos(theta) = sin(90° - theta)
  • Double angle: cos(2theta) = 2cos^2(theta) - 1 = cos^2(theta) - sin^2(theta)
  • Half angle: cos(theta/2) = +/- sqrt((1 + cos(theta)) / 2)
  • Sum formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
  • Difference formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. For any angle theta measured counterclockwise from the positive x-axis, the point on the unit circle is (cos(theta), sin(theta)). This geometric interpretation makes it easy to understand why cosine ranges between -1 and 1.

Applications of Cosine

  • Physics: Describing wave motion, oscillations, and electromagnetic waves.
  • Engineering: Signal processing, AC circuit analysis, and structural analysis.
  • Navigation: Computing distances using the law of cosines.
  • Computer graphics: Rotations, lighting calculations, and animation.
  • Music: Sound waves and harmonics are modeled using cosine functions.

Law of Cosines

The law of cosines generalizes the Pythagorean theorem: c^2 = a^2 + b^2 - 2ab*cos(C), where C is the angle opposite side c. This formula is used to find unknown sides or angles in any triangle, not just right triangles.