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Evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. \( \cot \pi \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cot \pi=\square \) B. The expression is undefined.

Ask by Chambers Craig. in the United States
Dec 03,2024

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B. The expression is undefined.

Solution

To evaluate \( \cot \pi \), we first recall the definition of the cotangent function: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] Now, we can find the values of \( \cos \pi \) and \( \sin \pi \): \[ \cos \pi = -1 \] \[ \sin \pi = 0 \] Substituting these values into the cotangent function gives: \[ \cot \pi = \frac{\cos \pi}{\sin \pi} = \frac{-1}{0} \] Since division by zero is undefined, we conclude that: \[ \cot \pi \text{ is undefined.} \] Thus, the correct choice is: B. The expression is undefined.

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The trigonometric function \( \cot \pi \) corresponds to the cotangent of the angle \( \pi \) radians, which is equivalent to \( 180^\circ \). At this angle, the sine value is \( 0 \), and cosine value is \( -1 \). Since cotangent is defined as \( \cot x = \frac{\cos x}{\sin x} \), we have \( \cot \pi = \frac{-1}{0} \), which is undefined. Therefore, the correct choice is: B. The expression is undefined.

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