For which value of \( \theta \) is \( \cot (\theta) \) undefined? \( 90^{\circ} \) \( 180^{\circ} \) \( 270^{\circ} \) \( 450^{\circ} \)

1. Recall that the cotangent function is defined as: \[ \cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)} \] 2. The cotangent will be undefined when the denominator is zero, i.e., when: \[ \sin(\theta)=0 \] 3. Evaluate the sine values for the given angles: - For \( 90^{\circ} \): \(\sin(90^{\circ})=1\) (defined) - For \( 180^{\circ} \): \(\sin(180^{\circ})=0\) (undefined) - For \( 270^{\circ} \): \(\sin(270^{\circ})=-1\) (defined) - For \( 450^{\circ} \): \(450^{\circ}-360^{\circ}=90^{\circ}\) so \(\sin(450^{\circ})=\sin(90^{\circ})=1\) (defined) 4. Thus, the only angle where \(\sin(\theta)=0\) among the options is: \[ 180^{\circ} \] The value of \(\theta\) for which \(\cot (\theta)\) is undefined is \(\theta = 180^{\circ}\).