If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. cot[n • 180°]

Verified step by step guidance

1. Recall the definition of the cotangent function: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
2. Substitute \(\theta = n \cdot 180^\circ\) into the cotangent function, so we consider \(\cot (n \cdot 180^\circ) = \frac{\cos (n \cdot 180^\circ)}{\sin (n \cdot 180^\circ)}\).
3. Evaluate \(\sin (n \cdot 180^\circ)\): since \(\sin\) of any integer multiple of \(180^\circ\) is zero, \(\sin (n \cdot 180^\circ) = 0\).
4. Evaluate \(\cos (n \cdot 180^\circ)\): \(\cos\) of integer multiples of \(180^\circ\) alternates between \(1\) and \(-1\), specifically \(\cos (n \cdot 180^\circ) = (-1)^n\).
5. Since the denominator \(\sin (n \cdot 180^\circ)\) is zero, the expression \(\cot (n \cdot 180^\circ)\) is undefined for all integer values of \(n\).

Key Concepts

Cotangent Function and Its Definition

The cotangent function, cot(θ), is defined as the ratio of the cosine to the sine of an angle θ, i.e., cot(θ) = cos(θ)/sin(θ). It is undefined where sin(θ) = 0, which occurs at integer multiples of 180°. Understanding this ratio is essential to evaluate cot[n • 180°].

Properties of Angles in Degrees and Multiples of 180°

Angles that are integer multiples of 180° correspond to points on the unit circle where the sine function is zero and cosine is either 1 or -1. Specifically, sin(n•180°) = 0 and cos(n•180°) = (-1)^n. This property helps determine the value or undefined nature of cot[n • 180°].

Undefined Values in Trigonometric Functions

A trigonometric function is undefined when its denominator is zero. For cotangent, this happens when sin(θ) = 0. Since sin(n•180°) = 0, cot[n • 180°] is undefined for all integer n. Recognizing when functions are undefined is crucial for correctly interpreting trigonometric expressions.