Question

if n is an integer, n·180° represents an integer multiple of 180° and (2n + 1)·90° represents an odd integer multiple of 90°, and so on. determine whether the expression is equal to 0, 1, −1, or is undefined. cot n·180° cotn·180° dropdown with options =1, =0, =−1, is undefined.

Explanation:

Step1: Recall cotangent definition

The cotangent function is defined as \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). For \(\theta = n\cdot180^{\circ}\), where \(n\) is an integer, we first find \(\sin(n\cdot180^{\circ})\) and \(\cos(n\cdot180^{\circ})\).

Step2: Evaluate sine and cosine

We know that \(\sin(k\cdot180^{\circ}) = 0\) for any integer \(k\) (since the sine function has a period of \(360^{\circ}\) and \(\sin(180^{\circ}) = 0\), \(\sin(360^{\circ})=0\), etc.). For \(\cos(n\cdot180^{\circ})\), when \(n\) is even, \(\cos(n\cdot180^{\circ}) = 1\); when \(n\) is odd, \(\cos(n\cdot180^{\circ})=- 1\), but regardless, \(\sin(n\cdot180^{\circ}) = 0\).

Step3: Analyze cotangent value

Since \(\cot(n\cdot180^{\circ})=\frac{\cos(n\cdot180^{\circ})}{\sin(n\cdot180^{\circ})}\) and the denominator \(\sin(n\cdot180^{\circ}) = 0\), division by zero is undefined in mathematics. So \(\cot(n\cdot180^{\circ})\) is undefined.

Answer:

is undefined.