Cosine Calculator (cos)
Calculate the cosine of any angle in degrees or radians. Shows unit circle coordinates, exact values for special angles, and step-by-step steps.
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What Is a Cosine Calculator (cos)?
In mathematics, the cosine is a fundamental trigonometric function. In a right triangle, the cosine of an acute angle is the ratio of the length of the adjacent leg to the hypotenuse: cos(θ) = Adjacent / Hypotenuse.
How to Use This Calculator
- Enter the angle θ and select Degrees or Radians.
- View the decimal cosine value and exact special angle representation.
- Examine the unit circle coordinates (cos θ, sin θ) and step-by-step derivation.
Cosine Trigonometric Definition
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = x_{\text{unit circle}}On the Cartesian unit circle of radius 1, cosine corresponds directly to the horizontal x-coordinate of the terminal point.
Worked Example
Scenario: Calculate cos(60°)
Given: θ = 60° = π / 3 radians
In a 30°-60°-90° triangle, the adjacent leg to 60° is half the hypotenuse.
cos(60°) = 1 / 2 = 0.5
Tips & Key Notes
- The range of cosine is strictly bounded between -1 and 1: -1 ≤ cos(θ) ≤ 1.
- Cosine is an even function: cos(-θ) = cos(θ).
- Pythagorean identity: sin²(θ) + cos²(θ) = 1 for all real angles θ.
Frequently Asked Questions
What are the exact values of cosine for common angles?
cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0.
What is the relationship between sine and cosine?
Sine and cosine are cofunctions: cos(θ) = sin(90° - θ). A cosine wave is a sine wave shifted 90° to the left.
What is the reciprocal of the cosine function?
The reciprocal of cosine is secant: sec(θ) = 1 / cos(θ).
Why is cos(90°) equal to zero?
At 90°, the terminal ray points vertically straight up along the y-axis, where the horizontal x-coordinate is 0.
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