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Understanding the Cosine Function
What is Cosine?
The cosine function (cos θ) is a fundamental trigonometric function that relates angles to the ratio of sides in a right triangle. In a right triangle:
cos θ = adjacent / hypotenuse
Unit Circle and Cosine
On the unit circle, cosine represents the x-coordinate of any point on the circle. This geometric interpretation helps us understand several key properties:
- Cosine values are always between -1 and 1
- Cosine is periodic with period 2π
- Cosine is an even function: cos(-θ) = cos(θ)
- At 0°, cosine reaches its maximum value of 1
- At 180°, cosine reaches its minimum value of -1
Important Properties
Domain
All real numbers (−∞ to +∞)
Range
[−1, 1]
Period
2π radians (360°)
Even Function
cos(−x) = cos(x)
Special Angles and Values
| Angle (degrees) | Angle (radians) | Cosine Value | Exact Form |
|---|---|---|---|
| 0° | 0 | 1 | 1 |
| 60° | π/3 | 0.5 | 1/2 |
| 90° | π/2 | 0 | 0 |
| 120° | 2π/3 | -0.5 | -1/2 |
| 180° | π | -1 | -1 |
Advanced Properties
Pythagorean Identity
sin²θ + cos²θ = 1
Double Angle Formula
cos(2θ) = cos²θ - sin²θ
Power Reduction
cos²θ = (1 + cos(2θ))/2
Cosine of Sum
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
Applications of Cosine Function
- Wave Motion Analysis
- Signal Processing
- Alternating Current
- Mechanical Vibrations
- Sound Wave Modeling
Real-World Applications
Physics
Used in projectile motion, simple harmonic motion, and wave propagation
Engineering
Essential in AC circuit analysis and mechanical systems
Navigation
Used in GPS systems and maritime navigation