I have a question thatasks me to find an algebraic

eiraszero11cu 2021-12-08 Answered
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 1x2
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Edward Patten
Answered 2021-12-09 Author has 38 answers
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=1x2
Then sin(y) is the ratio of the opposite size, z=1x2 and the hypotenuse, 1. We may now say sin(arccos(x))=1x2.
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esfloravaou
Answered 2021-12-10 Author has 43 answers
Set α=arccosx. Since, by definition, α[0,π], you know that sinα0, so
sinarccosx=sinα=1cos2α=1(cosarccosx)2=1x2
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