Evaluate the expression. sin − 1 ( 3 2 ) \sin^{-1}\left(\frac{\sqrt3}{2}\right) sin − 1 ( 2 3 <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"> )

Here we're asked to evaluate the expression the inverse sine of the square root of 3 over 2. So we can think of this as the sine of what angle is going to give me root 3 over 2 and we want to find the angle for which that is true. Now looking within my interval, negative pi over 2 to pi over 2, because I'm taking the inverse sine of a positive value, I know that my solution, my angle, has to be in quadrant 1. So of these angles in quadrant 1, which one gives me a sine of root 3 over 2? Now remember, sine is your y values when dealing with the unit circle, and here at pi over 3, I know that the sine is root 3 over 2, so that represents my solution. Pi over 3 and we are completely good to go here. Thanks for watching and let me know if you have questions.

0. Functions

Inverse Trigonometric Functions

Calculus

0. Functions

Inverse Trigonometric Functions

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Multiple Choice

Evaluate the expression.
$\sin^{-1}\left(\frac{\sqrt3}{2}\right)$

$pi over 6$

$3 pi over 2$

$pi over 3$

$- \frac{π}{3}$

Verified step by step guidance

Understand that the problem involves evaluating the inverse sine function, $ \sin^{-1} $, which gives the angle whose sine is the given value.

Recognize that $ \sin^{-1} \left( \frac{\sqrt{3}}{2} \right) $ asks for the angle whose sine is $ \frac{\sqrt{3}}{2} $.

Recall that the sine of $ \frac{\pi}{3} $ is $ \frac{\sqrt{3}}{2} $. Therefore, $ \sin^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{3} $.

Consider the range of the inverse sine function, which is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This means the angle $ \frac{\pi}{3} $ is valid within this range.

Conclude that the correct answer is $ \frac{\pi}{3} $, as it is the angle within the range of $ \sin^{-1} $ that corresponds to the given sine value.