What is arctan of 1?
Understand the Problem
The question is asking for the value of the arctangent function when the input is 1. In trigonometry, arctan(1) corresponds to the angle whose tangent is 1, which is a fundamental concept.
Answer
\( \frac{\pi}{4} \)
Answer for screen readers
The final answer is ( \frac{\pi}{4} )
Steps to Solve
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Understand what arctan(1) means
The arctangent function, denoted as $\arctan(x)$ or $\tan^{-1}(x)$, gives the angle whose tangent is $x$. So, $\arctan(1)$ is the angle whose tangent is 1.
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Recall the fundamental angle with tangent equal to 1
From trigonometry, we know that the tangent of $45^\circ$ (or $\frac{\pi}{4}$ radians) is equal to 1.
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Use the principal value range of the arctangent function
The principal value of the arctangent function usually lies within the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$. Since $\frac{\pi}{4}$ is within this interval, that is our answer.
Therefore, $\arctan(1) = \frac{\pi}{4}$.
The final answer is ( \frac{\pi}{4} )
More Information
In trigonometry, the tangent of a 45-degree angle, or ( \frac{\pi}{4} ) radians, is 1. Hence, the arctangent of 1 is ( \frac{\pi}{4} ).
Tips
A common mistake is not considering the principal value range of the arctangent function, which is between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.
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