Cosine Calculator
Calculate cos(x) for any angle or find the angle from a cosine value using arccos. Features unit circle and cosine wave visualizations.
Result
cos(60)
0.500000
Exact: 1/2
Calculation Mode
Input
Enter angle in degrees
Unit Circle Visualization
On the unit circle, cos(60.0) is the x-coordinate of the point: 0.500000
Related Trigonometric Values
cos(60)
0.500000
sin(60)
0.866025
tan(60)
1.732051
sec(60) = 1/cos
2.000000
Cofunction Identity: cos(60) = sin(30)
Pythagorean Identity: sin^2(60) + cos^2(60) = 1.000000 ≈ 1
Cosine Wave Graph
Amplitude
1
Period
2pi
Phase Shift
0
Vertical Shift
0
Equation:
y = cos((x))
Special Cosine Values
| Angle (deg) | Radians | cos | Exact Value |
|---|---|---|---|
| 0 | 0 | 1.0000 | 1 |
| 30 | pi/6 | 0.8660 | sqrt(3)/2 |
| 45 | pi/4 | 0.7071 | sqrt(2)/2 |
| 60 | pi/3 | 0.5000 | 1/2 |
| 90 | pi/2 | 0.0000 | 0 |
| 120 | 2pi/3 | -0.5000 | -1/2 |
| 135 | 3pi/4 | -0.7071 | -sqrt(2)/2 |
| 150 | 5pi/6 | -0.8660 | -sqrt(3)/2 |
| 180 | pi | -1.0000 | -1 |
Cosine Function Properties
Domain
All real numbers (-inf, inf)
Range
[-1, 1]
Period
360 degrees (2pi radians)
Odd/Even
Even function: cos(-x) = cos(x)
Zeros
x = 90 + n*180, where n is integer
Maximum/Minimum
Max = 1 at 0, Min = -1 at 180
Cosine vs Sine Phase Relationship
Cosine is the sine function shifted 90 degrees to the left:
cos(x) = sin(x + 90)
sin(x) = cos(x - 90)
cos(x) = sin(90 - x)
This means the cosine wave leads the sine wave by a quarter period.
Result
cos(60)
0.500000
Try These Examples
Quick-start with common scenarios
Practice Problems
Test your skills with practice problems
Practice with 3 problems to test your understanding.
?What is Cosine?
Cosine (cos) is a trigonometric function that gives the ratio of the adjacent side to the hypotenuse in a right triangle: cos(angle) = adjacent/hypotenuse. On the unit circle, cosine equals the x-coordinate. Common values: cos(0)=1, cos(30)=sqrt(3)/2, cos(45)=sqrt(2)/2, cos(60)=0.5, cos(90)=0. The inverse cosine (arccos) finds the angle from a cosine value.
About the Cosine Function
The cosine function is one of the fundamental trigonometric functions. For an angle in a right triangle, cosine equals the length of the adjacent side divided by the hypotenuse. In the unit circle, cosine represents the x-coordinate of a point on the circle. Cosine oscillates between -1 and 1 and is essential in physics, engineering, and signal processing.
Key Facts
- cos(x) = adjacent / hypotenuse in a right triangle
- On the unit circle, cos(x) = x-coordinate of the point
- Range: cosine values are always between -1 and 1
- Period: cosine repeats every 360 degrees (2pi radians)
- cos(0) = 1, cos(90) = 0, cos(180) = -1, cos(270) = 0
- arccos(x) returns angle in range [0, 180] or [0, pi]
- Cosine is positive in quadrants I and IV (0-90, 270-360)
- cos(x) = sin(90 - x) - cofunction identity
Frequently Asked Questions
Cosine is a trigonometric function that, for an angle in a right triangle, equals the ratio of the adjacent side to the hypotenuse. On the unit circle, cos(angle) equals the x-coordinate of the point at that angle. Cosine oscillates between -1 and 1.
cos(60) = 0.5 or 1/2. This is one of the special angles. The exact value comes from the 30-60-90 triangle where the side adjacent to 60 is half the hypotenuse.
Arccos (also written as cos inverse or acos) returns the angle whose cosine equals a given value. For example, arccos(0.5) = 60 because cos(60) = 0.5. The result is always between 0 and 180 degrees.
Cosine is positive in quadrants I and IV (0-90 and 270-360 degrees) where the x-coordinate is positive. Cosine is negative in quadrants II and III (90-270 degrees) where the x-coordinate is negative.
Cosine and sine are cofunctions: cos(x) = sin(90-x) and sin(x) = cos(90-x). They also satisfy the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Cosine is the sine function shifted by 90 degrees.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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Result
cos(60)
0.500000
Exact: 1/2