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Which is equivalent to [tex]\sin ^{-1}\left(\tan \left(\frac{\pi}{4}\right)\right)[/tex] ? Give your answer in radians.

A. 0
B. [tex]\frac{\pi}{2}[/tex]
C. [tex]\pi[/tex]
D. [tex]2 \pi[/tex]


Sagot :

To solve the problem [tex]\(\sin^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right)\)[/tex], we need to evaluate it step-by-step.

First, let's evaluate the inner function [tex]\(\tan\left(\frac{\pi}{4}\right)\)[/tex].

Recall that:
[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]

Next, we need to evaluate the inverse sine (arcsine) of [tex]\(1\)[/tex]:
[tex]\[ \sin^{-1}(1) \][/tex]

By definition, [tex]\(\sin^{-1}(x)\)[/tex] is the value [tex]\(y\)[/tex] such that [tex]\(\sin(y) = x\)[/tex] and [tex]\(y\)[/tex] lies in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].

The value of [tex]\(y\)[/tex] that satisfies [tex]\(\sin(y) = 1\)[/tex] within this range is:
[tex]\[ \frac{\pi}{2} \][/tex]

Thus:
[tex]\[ \sin^{-1}(1) = \frac{\pi}{2} \][/tex]

Therefore:
[tex]\[ \sin^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right) = \frac{\pi}{2} \][/tex]

Given the possible options:
[tex]\[ 0, \frac{\pi}{2}, \pi, 2\pi \][/tex]

The correct one is:
[tex]\[ \boxed{\frac{\pi}{2}} \][/tex]
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