I have a question thatasks me to find an algebraic expression for sin⁡(arccos⁡(x)). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not even sure where to start here. It's supposed to be 1−x2

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I have a question thatasks me to find an algebraic expression for sin⁡(arccos⁡(x)). From the lone example in the book I seen they&#039;re doing some multistep thing with the identities, but I&#039;m just not even sure where to start here. It&#039;s supposed to be 1−x2

Edward Patten · Accepted Answer

What is arccos⁡(x)? it is the angle given by the ratio of sides of length x and 1. That is, on a triangle with adjacent side length x and hypotenuse length 1 we will find the angle arccos⁡(x).  Now what is sin⁡(y)? It is the ratio of the opposite side to the hypotenuse of a triangle with angle y in the respective part. So if we use y=arccos⁡(x), we have that sin⁡(y) is the ratio of the sides with degree given by a triangle whose adjacent and hypotenuse sides are length x and 1 respectively.  Since we have the adjacent and hypotenuse lengths, we can calculate the opposite sides length by the Pythagorean theorem. This means, if we use z for the opposite side, that z2+x2=12=1. Solving for z gives z=1−x2  Then sin⁡(y) is the ratio of the opposite size, z=1−x2 and the hypotenuse, 1. We may now say sin⁡(arccos⁡(x))=1−x2.

esfloravaou · Accepted Answer

Set α=arccos⁡x. Since, by definition, α∈[0,π], you know that sin⁡α≥0, so  sin⁡arccos⁡x=sin⁡α=1−cos2α=1−(cos⁡arccos⁡x)2=1−x2

I have a question thatasks me to find an algebraic

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