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 quadrant 1 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.

0. Functions

Inverse Trigonometric Functions

Calculus

0. Functions

Inverse Trigonometric Functions

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Multiple Choice

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

$pi over 4$

$- \frac{π}{4}$

$3 pi over 4$

$7 pi over 4$

Verified step by step guidance

Understand that sin⁡−1(x) or arcsin(x) is the inverse function of sin(x), which means it gives the angle whose sine is x.

Recognize that the expression sin⁡−1(√2/2) asks for the angle θ such that sin(θ) = √2/2.

Recall that the sine of π/4 (or 45 degrees) is √2/2. Therefore, one possible angle is θ = π/4.

Consider the range of the arcsin function, which is [-π/2, π/2]. This means the principal value of arcsin(√2/2) is within this range.

Conclude that the angle θ = π/4 is the correct answer since it lies within the range of the arcsin function and satisfies sin(θ) = √2/2.

4:49m

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Evaluate the expression.sin−1(22)\sin^{-1}\left(\frac{\sqrt2}{2}\right)sin−1(22)

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Inverse Trigonometric Functions practice set

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