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Calculate the cosine of any angle in degrees or radians. Cosine represents the x-coordinate on the unit circle and is fundamental to trigonometry.
| Degrees | Radians | Exact Value | Decimal |
|---|---|---|---|
| 0° | 0 | 1 | 1.0000 |
| 30° | π/6 | √3/2 | 0.8660 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/3 | 1/2 | 0.5000 |
| 90° | π/2 | 0 | 0.0000 |
| 120° | 2π/3 | -1/2 | -0.5000 |
| 135° | 3π/4 | -√2/2 | -0.7071 |
| 150° | 5π/6 | -√3/2 | -0.8660 |
| 180° | π | -1 | -1.0000 |
| 270° | 3π/2 | 0 | -0.0000 |
| 360° | 2π | 1 | 1.0000 |
Domain: All real numbers (-∞, ∞)
Range: [-1, 1]
Period: 2π radians (360°)
cos(θ + 2π) = cos(θ)
Even function: cos(-θ) = cos(θ)
y-axis symmetry
Zeros: 90°, 270°, ...
Max: 0°, 360° | Min: 180°
The cosine function is one of the fundamental trigonometric functions, closely related to the sine function. It originated from the study of right triangles and was developed by ancient mathematicians studying astronomy and geometry.
In a right triangle, the cosine of an angle θ equals the ratio of the length of the side adjacent to the angle to the length of the hypotenuse: cos(θ) = adjacent / hypotenuse.
On the unit circle, the cosine of an angle equals the x-coordinate of the point where the terminal side of the angle intersects the circle. This definition makes cosine applicable to all real numbers, not just acute angles.
Cosine values can be calculated using the unit circle definition or trigonometric identities. Memorizing common values speeds up calculations.
Step 1: Convert to radians if needed: 60° = π/3 rad
Step 2: Recall that cos(60°) = 1/2 (exact value)
Result: cos(60°) = 0.5
Step 1: Convert to radians: 45° = π/4 rad
Step 2: Recall that cos(45°) = √2/2 (exact value)
Result: cos(45°) ≈ 0.7071
Step 1: 120° is in Quadrant II
Step 2: Reference angle = 180° - 120° = 60°
Step 3: Cosine is negative in Quadrant II
Result: cos(120°) = -cos(60°) = -1/2 = -0.5
Cosine is a trigonometric function that returns the x-coordinate of a point on the unit circle. In a right triangle, cos(θ) = adjacent/hypotenuse.
Because cos(-θ) = cos(θ). Negating the angle reflects the point across the x-axis, which doesn't change the x-coordinate.
Cosine is positive in Quadrants I and IV (0° to 90°, 270° to 360°), and negative in Quadrants II and III (90° to 270°).
The range of cosine is [-1, 1]. The cosine of any angle will always be between -1 and 1, inclusive.
The period of cosine is 2π radians (360°). This means cos(θ + 2π) = cos(θ) for all values of θ.
Cosine and sine are complementary functions: cos(θ) = sin(90° - θ). They are also phase-shifted: cos(θ) = sin(θ + 90°).
Cosine equals zero at 90°, 270°, and all odd multiples of 90° (or π/2, 3π/2, ... in radians).
Cosine is used to calculate horizontal components of vectors, work done by forces, and to model wave phenomena in optics and acoustics.