Understanding the Cotangent Function

What is the Cotangent Function?

The cotangent function, denoted as cot(θ), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In a right-angled triangle, the cotangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side.

Formula and Definition

For a right-angled triangle with an angle θ, the cotangent function is defined as:

\[\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}\]

Where:

• "adjacent" is the length of the side adjacent to the angle θ
• "opposite" is the length of the side opposite to the angle θ

Calculation Steps

1. Identify the angle θ in degrees or radians.
2. If the angle is in degrees, convert it to radians using the formula: θ (in radians) = θ (in degrees) × π/180°.
3. Apply the cotangent function: cot(θ) = 1 / tan(θ).
4. Round the result to the desired number of decimal places.

Example Calculation

Let's calculate cot(60°):

1. θ = 60°
2. Convert to radians: 60° × π/180° = π/3 radians
3. cot(π/3) = 1 / tan(π/3) = 1 / √3 ≈ 0.5774

Visual Representation

This diagram illustrates cot(60°) in the unit circle. The green line represents the cotangent value, which is the x-coordinate (approximately 0.5774) of the point where a line perpendicular to the y-axis at y=1 intersects the angle's terminal side extended.