Cosine Calculator
Calculate cos(x) for any angle
Common Cosine Values
0°
1
30°
√3/2 ≈ 0.866
45°
√2/2 ≈ 0.707
60°
0.5 (1/2)
90°
0
180°
-1
270°
0
360°
1
About 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 adjacent side to the hypotenuse.
Definition
For a right triangle: cos(θ) = adjacent / hypotenuse
On the unit circle: cos(θ) = x-coordinate of the point on the circle
Properties
- Range: −1 ≤ cos(θ) ≤ 1
- Period: 360° (2π radians)
- cos(0°) = 1, cos(90°) = 0
- cos(θ + 360°) = cos(θ)
- cos(−θ) = cos(θ) (even function)
Applications
- Physics: Projectile motion, forces
- Engineering: Mechanical systems, vibrations
- Computer Graphics: Rotations, transformations
- Navigation: GPS, bearing calculations
- Astronomy: Celestial mechanics
How to Use This Calculator
Enter the Angle
Input the angle value you want to calculate the cosine for. You can use degrees or radians.
Select Unit
Choose whether your angle is in degrees (°) or radians (rad).
Calculate
Click the "Calculate Cosine" button to get the result instantly.
Review Results
View the cosine value along with the angle in both degrees and radians, plus related trigonometric values.
Formula
cos(θ) = adjacent / hypotenuse
In a right triangle
Unit Circle Definition:
cos(θ) = x-coordinate on unit circle
Key Relationships:
- cos²(θ) + sin²(θ) = 1 (Pythagorean identity)
- cos(θ) = sin(90° − θ)
- cos(−θ) = cos(θ) (even function)
Common Values:
cos(0°) = 1
cos(30°) = √3/2
cos(45°) = √2/2
cos(60°) = 1/2
cos(90°) = 0
About Cosine Calculator
The Cosine Calculator is a powerful tool for calculating cosine values for any angle in degrees or radians. Cosine is one of the fundamental trigonometric functions that appears in many areas of mathematics, physics, and engineering. This calculator helps you quickly and accurately determine cosine values without manual computation.
What is Cosine?
Cosine (cos) is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In the context of the unit circle, cosine represents the x-coordinate of a point on the circle.
When to Use This Calculator
- Trigonometry Problems: Solve trigonometric equations and identities
- Physics Calculations: Calculate components of forces, velocities, and accelerations
- Engineering: Design mechanical systems, analyze vibrations and oscillations
- Navigation: Compute distances, bearings, and GPS coordinates
- Computer Graphics: Perform rotations and transformations
- Verification: Double-check manual calculations and homework
Why Use Our Calculator?
- ✅ Instant Results: Get accurate cosine values immediately
- ✅ Multiple Units: Supports both degrees and radians
- ✅ Related Values: Shows sine and tangent for the same angle
- ✅ Easy to Use: Simple interface for all skill levels
- ✅ 100% Free: No registration or payment required
- ✅ Mobile Friendly: Works on all devices and screen sizes
- ✅ Accurate: Precise mathematical calculations with high precision
Properties of Cosine
- Range: The cosine function outputs values between -1 and 1
- Period: Cosine is periodic with period 360° (2π radians)
- Even Function: cos(−θ) = cos(θ), making it symmetric about the y-axis
- Domain: All real numbers (angles can be any value)
Frequently Asked Questions
What is cos(0°)?
cos(0°) = 1. This is the maximum value of the cosine function. At 0°, the point on the unit circle is at (1, 0), so the x-coordinate (cosine) is 1.
How is cosine related to sine?
The most fundamental relationship is the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Also, cos(θ) = sin(90° − θ), meaning cosine and sine are complementary functions.
Why does cosine range from -1 to 1?
On the unit circle (radius = 1), cosine represents the x-coordinate of a point on the circle. The x-coordinate ranges from -1 (leftmost point) to +1 (rightmost point) as the angle rotates around the circle.
What is the difference between cos and arccos?
cos finds the ratio (cosine value) given an angle, while arccos (inverse cosine) finds the angle given a cosine value. They are inverse operations: if cos(θ) = x, then arccos(x) = θ.
What are the cosine values I should memorize?
Key values to memorize: cos(0°)=1, cos(30°)=√3/2≈0.866, cos(45°)=√2/2≈0.707, cos(60°)=1/2=0.5, cos(90°)=0. These are essential for quick calculations and problem-solving!
Can cosine be negative?
Yes! Cosine is negative in the second quadrant (90°-180°) and third quadrant (180°-270°). It's positive in the first quadrant (0°-90°) and fourth quadrant (270°-360°).
How do I convert between degrees and radians?
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. This calculator handles both units automatically.