tan inverse of infinity
Understand the Problem
The question is asking for the value of the inverse tangent (arctan) function when the input is infinity. This relates to understanding limits and the behavior of the arctan function as its argument approaches infinity.
Answer
\frac{\pi}{2}
Answer for screen readers
The final answer is \frac{\pi}{2}
Steps to Solve
- Recognize the relationship of the arctan function and infinity
The arctan (inverse tangent) function provides the angle whose tangent value is the input. We need to determine the angle whose tangent value approaches infinity.
- Understand the tangent function behavior
The tangent function, $\tan(x)$, has the property that as $x$ approaches $\frac{\pi}{2}$ from the left, $\tan(x)$ approaches positive infinity. Mathematically, $$ \lim_{x \to \frac{\pi}{2}^-} \tan(x) = +\infty $$
- Determine the angle for $\arctan(\infty)$
Thus, if $\tan(\frac{\pi}{2})$ is undefined because it asymptotically approaches infinity, we can infer that: $$ \arctan(\infty) = \frac{\pi}{2} $$
The final answer is \frac{\pi}{2}
More Information
The value of the arctan (inverse tangent) function for infinity is $\frac{\pi}{2}$ radians, which is equivalent to 90 degrees.
Tips
Common mistakes include confusing the behavior of the tangent function with other trigonometric functions or assuming the limits apply at finite values instead of approaching them at infinity.
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