\begin{tabular}{l} If $$ n $$ is an integer, $$ n \cdot 180^{\circ} $$ represents an integer multiple of $$ 180^{\circ} $$ and $$ (2 n+1) \cdot 90^{\circ} $$ represents an odd integer multiple of $$ 90^{\circ} $$, and so on, Determine whether the expression is equal to $$ 0,1,-1 $$, or is \ undefined. \ $$ an \left[(2 n+1) \cdot 90^{\circ} ight] $$ \ $$ an \left[(2 n+1) \cdot 90^{\circ} ight] $$ \ \hline\end{tabular}

Question

\begin{tabular}{l} If $$ n $$ is an integer, $$ n \cdot 180^{\circ} $$ represents an integer multiple of $$ 180^{\circ} $$ and $$ (2 n+1) \cdot 90^{\circ} $$ represents an odd integer multiple of $$ 90^{\circ} $$, and so on, Determine whether the expression is equal to $$ 0,1,-1 $$, or is \ undefined. \ $$ 	an \left[(2 n+1) \cdot 90^{\circ}ight] $$ \ $$ 	an \left[(2 n+1) \cdot 90^{\circ}ight] $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To evaluate the expression $$ 	an \left[(2n + 1) \cdot 90^{\circ}ight] $$ where $$ n $$ is an integer, let's analyze the angles involved:

1. **Expression Breakdown:**
   - $$ (2n + 1) \cdot 90^{\circ} $$ represents an odd multiple of $$ 90^{\circ} $$.
   - For example:
     - If $$ n = 0 $$, angle = $$ 90^{\circ} $$
     - If $$ n = 1 $$, angle = $$ 270^{\circ} $$
     - If $$ n = -1 $$, angle = $$ -90^{\circ} $$ (which is equivalent to $$ 270^{\circ} $$ in standard position)

2. **Tangent of Odd Multiples of $$ 90^{\circ} $$:**
   - $$ 	an(90^{\circ}) $$ is **undefined** because it represents a vertical line where the tangent function approaches infinity.
   - Similarly, $$ 	an(270^{\circ}) $$, $$ 	an(450^{\circ}) $$, etc., are also **undefined** for the same reason.

3. **Conclusion:**
   - Regardless of the integer value of $$ n $$, $$ 	an \left[(2n + 1) \cdot 90^{\circ}ight] $$ does not yield a finite value.
   - Therefore, the expression is **undefined**.

**Answer:**  
**Undefined**
The expression $$ 	an \left[(2n + 1) \cdot 90^{\circ}ight] $$ is **undefined** for any integer $$ n $$.

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