Cotangent Calculator

Definition and Formula of Cotangent

The cotangent function is a fundamental trigonometric function, often denoted as \(\cot(\theta)\), where \(\theta\) represents an angle, typically measured in radians.

In a right triangle, the cotangent of an angle \(\theta\) is defined as the ratio of the adjacent side to the opposite side: \( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{b}{a} \) This means the cotangent represents the ratio of the horizontal to the vertical component of an angle.

On the unit circle, the cotangent is defined as the ratio of the \(x\)-coordinate to the \(y\)-coordinate of the point corresponding to angle \(\theta\): \( \cot(\theta) = \frac{x}{y} \) Here, \(x\) and \(y\) are the coordinates of the point on the unit circle.

Examples

Example 1: Calculating Cotangent in a Right Triangle

A right triangle has an acute angle \(\theta = 45^\circ\). The lengths of the opposite and adjacent sides are both 6. Find the cotangent value of this angle.

Example 2: Real-World Application of Cotangent

You are observing a lighthouse that is 50 meters tall, and you are standing 80 meters away from its base. Calculate the angle \(\theta\) between you and the lighthouse.

Cotangent Graph and Properties

The graph of the cotangent function displays periodic oscillations with vertical asymptotes in each period. Key characteristics include:

• Periodicity: The cotangent function has a period of \(\pi\) (180°), meaning it repeats every \(\pi\) radians.
• Monotonicity: Within each period, cotangent decreases monotonically.
• Even Function: \(\cot(-\theta) = \cot(\theta)\), indicating symmetry about the \(y\)-axis.
• Amplitude: Cotangent has no amplitude because its range extends from \(-\infty\) to \(\infty\).
• Vertical Asymptotes: Vertical asymptotes occur at \(\theta = n\pi\) (where \(n\) is an integer).
• Domain and Range: The domain of the cotangent function is all real angles except multiples of \(\pi\), and the range is \(\mathbb{R}\) (all real numbers).

Cotangent in Different Quadrants

The behavior of the cotangent function across quadrants is as follows:

Additional Cotangent Calculations

1. Reciprocal of Cotangent (Tangent Function)

The reciprocal of the cotangent function is the tangent function: \( \frac{1}{\cot(\theta)} = \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \) Cotangent is undefined when \(\tan(\theta) = 0\).

2. Derivative of Cotangent

The derivative of cotangent is negative secant squared: \( \frac{d}{d\theta} \cot(\theta) = -\sec^2(\theta) \)

3. Integral of Cotangent

The integral of the cotangent function is: \( \int \cot(\theta) \, d\theta = \ln|\sin(\theta)| + C \)

4. Arccotangent (Inverse Cotangent)

The arccotangent function (\(\text{arccot}(x)\)) calculates the angle corresponding to a given cotangent value: \( \theta = \text{arccot}(x) \)

Common Cotangent Values Table