Understanding Undefined Trigonometric Functions at 0 Degrees

The question asks us to identify which among the given trigonometric function values is not defined. We need to evaluate each option at the specified angle, which is 0 degrees.

Evaluating Trigonometric Functions at 0 Degrees

Let's consider each option:

• Option 1: $\sin 0^\circ$

  The value of sine at 0 degrees is 0.

  $\sin 0^\circ = 0$

  This value is clearly defined.

• Option 2: $\cos 0^\circ$

  The value of cosine at 0 degrees is 1.

  $\cos 0^\circ = 1$

  This value is also defined.

• Option 3: $\tan 0^\circ$

  The tangent function is defined as the ratio of sine to cosine, i.e., $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

  For $\theta = 0^\circ$, we have:

  $\tan 0^\circ = \frac{\sin 0^\circ}{\cos 0^\circ} = \frac{0}{1} = 0$

  This value is defined.

• Option 4: $\operatorname{cosec} 0^\circ$

  The cosecant function is the reciprocal of the sine function, i.e., $\operatorname{cosec} \theta = \frac{1}{\sin \theta}$.

  For $\theta = 0^\circ$, we have:

  $\operatorname{cosec} 0^\circ = \frac{1}{\sin 0^\circ} = \frac{1}{0}$

  Division by zero is an operation that is undefined in mathematics. Therefore, $\operatorname{cosec} 0^\circ$ is not defined.

Summary of Values at 0 Degrees

Based on the evaluations, $\operatorname{cosec} 0^\circ$ is the only value among the options that is not defined because it involves division by zero.