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°]

검증된 단계별 안내

Recall the definition of the cotangent function: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Substitute \(\theta = n \cdot 180^\circ\) into the cotangent function, so we consider \(\cot (n \cdot 180^\circ) = \frac{\cos (n \cdot 180^\circ)}{\sin (n \cdot 180^\circ)}\).

Evaluate \(\sin (n \cdot 180^\circ)\): since \(\sin\) of any integer multiple of \(180^\circ\) is zero, \(\sin (n \cdot 180^\circ) = 0\).

Evaluate \(\cos (n \cdot 180^\circ)\): \(\cos\) of integer multiples of \(180^\circ\) alternates between \(1\) and \(-1\), specifically \(\cos (n \cdot 180^\circ) = (-1)^n\).

Since the denominator \(\sin (n \cdot 180^\circ)\) is zero, the expression \(\cot (n \cdot 180^\circ)\) is undefined for all integer values of \(n\).

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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.