Evaluate the expression. sin − 1 ( 2 2 ) \sin^{-1}\left(\frac{\sqrt2}{2}\right) sin − 1 ( 2 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> )

this problem we're asked to evaluate the expression the inverse sine of root 2 over 2. So we're looking for an angle for which the sine is equal to root 2 over 2. Now because this is a positive value and I know that my angle has to be in between negative pi over 2 and positive pi over 2, I know that it has to be in a quadrant one specifically where all of my sine values are positive. So in that first quadrant, where is my sine equal to root 2 over 2? Well for pi over 4 I know that my sine is root 2 over 2 so this represents my solution pi over 4 and we're done here. Thanks for watching and I'll see you in the next one.

Evaluate the expression. sin ⁡ − 1 ( 2 2 ) \sin^{-1}\left(\frac{\sqrt2}{2}\right) sin − 1 ( 2 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> )

− π 4 -\frac{\pi}{4} − 4 π

11. Inverse Trigonometric Functions and Basic Trig Equations

Inverse Sine, Cosine, & Tangent

Precalculus

11. Inverse Trigonometric Functions and Basic Trig Equations

Inverse Sine, Cosine, & Tangent

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Multiple Choice

Evaluate the expression.
$\sin^{-1}\left(\frac{\sqrt2}{2}\right)$

$\frac{\pi}{4}$

$-\frac{\pi}{4}$

$\frac{3\pi}{4}$

$\frac{7\pi}{4}$

Verified step by step guidance

Understand that the expression $ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) $ asks for the angle whose sine is $ \frac{\sqrt{2}}{2} $.

Recall that the sine of $ \frac{\pi}{4} $ is $ \frac{\sqrt{2}}{2} $. Therefore, $ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4} $.

Consider the range of the inverse sine function, which is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This means the principal value of $ \sin^{-1} $ will be within this interval.

Verify that $ \frac{\pi}{4} $ is within the range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$, confirming it as a valid solution.

Conclude that the correct angle corresponding to $ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) $ is $ \frac{\pi}{4} $.