Select the correct choice that completes the sentence below. If a quadrantal angle \( \theta \) is coterminal with \( 0^{\circ} \) or \( 180^{\circ} \), then the trigonometric functions are undefined ,

If a quadrantal angle \( \theta \) is coterminal with \( 0^{\circ} \) or \( 180^{\circ} \), then the **cosecant** and **cotangent** trigonometric functions are undefined. ### Explanation: - **Cosecant (\( \csc \theta \))** is defined as the reciprocal of the sine function: \[ \csc \theta = \frac{1}{\sin \theta} \] At \( 0^{\circ} \) and \( 180^{\circ} \), \( \sin \theta = 0 \), making \( \csc \theta \) undefined. - **Cotangent (\( \cot \theta \))** is defined as the reciprocal of the tangent function or the ratio of cosine to sine: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} \] Similarly, since \( \sin \theta = 0 \) at these angles, \( \cot \theta \) is also undefined. Other trigonometric functions like sine, cosine, tangent, and secant remain defined at these quadrantal angles: - **Sine (\( \sin \theta \))** is \( 0 \) at both \( 0^{\circ} \) and \( 180^{\circ} \), which is defined. - **Cosine (\( \cos \theta \))** is \( 1 \) at \( 0^{\circ} \) and \( -1 \) at \( 180^{\circ} \), both of which are defined. - **Tangent (\( \tan \theta \))** is \( 0 \) at both angles, also defined. - **Secant (\( \sec \theta \))** is the reciprocal of cosine, so it is \( 1 \) at \( 0^{\circ} \) and \( -1 \) at \( 180^{\circ} \), both defined. ### Summary: - **Undefined Functions at \( 0^{\circ} \) and \( 180^{\circ} \):** - Cosecant (\( \csc \theta \)) - Cotangent (\( \cot \theta \)) Therefore, the completed sentence is: *If a quadrantal angle \( \theta \) is coterminal with \( 0^{\circ} \) or \( 180^{\circ} \), then the **cosecant** and **cotangent** trigonometric functions are undefined.*