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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]
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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