Cosine Triangle Calculator
Calculate cosine values, angles, and triangle properties with step-by-step solutions
Cosine Calculator
Results
Exact Value: cos(90°) = 0
Step-by-Step Solution
1. Given angle: 0° (degrees)
2. Convert to radians: 0° × (π/180) = 0.0000 radians
3. Apply cosine function: cos(0.0000) = 0.000000
4. Exact value: cos(90°) = 0
Example Calculations
30° Angle
Given: θ = 30°
Calculation: cos(30°)
Result: cos(30°) = √3/2 ≈ 0.866
Exact value: √3/2
Right Triangle
Given: Adjacent = 3, Hypotenuse = 5
Calculation: cos(θ) = 3/5
Result: cos(θ) = 0.6
Angle: θ = arccos(0.6) ≈ 53.13°
Common Cosine Values
Cosine Properties
Range: -1 ≤ cos(θ) ≤ 1
Period: 360° (2π radians)
Even function: cos(-θ) = cos(θ)
cos(θ) = adjacent/hypotenuse
Decreasing in [0°, 180°]
Quick Tips
Cosine is positive in quadrants I and IV
Cosine is negative in quadrants II and III
Use SOHCAHTOA: Cosine = Adjacent/Hypotenuse
cos²(θ) + sin²(θ) = 1 (Pythagorean identity)
Understanding the Cosine Function
What is Cosine?
The cosine function is one of the fundamental trigonometric functions. In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. It's a key component in trigonometry and has applications across mathematics, physics, and engineering.
Right Triangle Definition
cos(θ) = adjacent / hypotenuse
where θ is the angle in question
Key Applications
- •Calculating unknown sides and angles in triangles
- •Analyzing periodic functions and waves
- •Engineering and physics calculations
- •Computer graphics and 3D transformations
Unit Circle
On the unit circle, cosine represents the x-coordinate of a point. This extends the definition beyond right triangles and allows cosine to be defined for any angle, including angles greater than 90° and negative angles.
Special Right Triangles
30-60-90 Triangle
Sides in ratio 1 : √3 : 2
cos(30°) = √3/2 ≈ 0.866
cos(60°) = 1/2 = 0.5
45-45-90 Triangle
Sides in ratio 1 : 1 : √2
cos(45°) = √2/2 ≈ 0.707
cos(45°) = sin(45°)