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

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

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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.*
The cosecant and cotangent trigonometric functions are undefined when a quadrantal angle $$ \theta $$ is coterminal with $$ 0^{\circ} $$ or $$ 180^{\circ} $$.

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 ,

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