To find the value of [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex], we need to understand the behavior of the tangent function near [tex]\(\frac{\pi}{2}\)[/tex].
1. Understanding the Tangent Function: The tangent function, [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex], is calculated by dividing the sine of an angle by the cosine of the same angle.
2. Behavior Near [tex]\(\frac{\pi}{2}\)[/tex]: As [tex]\(x\)[/tex] approaches [tex]\(\frac{\pi}{2}\)[/tex], the cosine of [tex]\(x\)[/tex] ([tex]\(\cos(\frac{\pi}{2})\)[/tex]) approaches 0. In mathematical terms:
- [tex]\(\sin\left(\frac{\pi}{2}\right) = 1\)[/tex]
- [tex]\(\cos\left(\frac{\pi}{2}\right) = 0\)[/tex]
3. Division by Zero: When the cosine of an angle approaches 0, the value of [tex]\(\tan(x)\)[/tex] tends to become very large, because dividing by a number close to zero results in a very large value. However, at exactly [tex]\(\frac{\pi}{2}\)[/tex], division by zero would occur, which is typically considered undefined in basic trigonometric functions.
4. Evaluating [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex]: Despite the general expectation of an undefined result due to division by zero, the actual computed value of [tex]\(\tan\left(\frac{\pi}{2}\right)\)[/tex] here provides a large numerical value:
[tex]\[
\tan\left(\frac{\pi}{2}\right) \approx 1.633123935319537 \times 10^{16}
\][/tex]
This output indicates an extremely large finite value, closely aligned with the behavior of the tangent function as it approaches [tex]\(\frac{\pi}{2}\)[/tex]. At the very point though, it’s more accurate to say the value is undefined in a pure theoretical sense.
Hence, evaluating the choices given:
A. 0 — Incorrect.
B. 1 — Incorrect.
C. -1 — Incorrect.
D. Undefined — The most accurate theoretical description, given it is close to an undefined condition.
Correct Answer: D. Undefined
