I have a question that asks me khổng lồ 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 khổng lồ start here. It"s supposed to lớn be \$sqrt1-x^2\$

Set \$alpha=arccos x\$. Since, by definition, \$alphain<0,pi>\$, you know that \$sinalphage0\$, so\$\$sinarccos x=sinalpha=sqrt1-cos^2alpha=sqrt1-(cosarccos x)^2=sqrt1-x^2\$\$

Let \$arccos(x)= heta\$

\$x=cos( heta)\$

You require \$y=sin(arccos(x))=sin( heta)\$

Identity: \$cos^2( heta)+sin^2( heta)=1\$

\$x^2+y^2=1\$

What is \$arccos(x)\$? it is the angle given by the ratio of sides of length \$x\$ và \$1\$. That is, on a triangle with adjacent side length \$x\$ and hypotenuse length \$1\$ we will find the angle \$arccos(x)\$.

(There is a picture for this which I hope you can get from what I describe).

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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\$ và \$1\$ respectively.

Since we have the adjacent & hypotenuse lengths, we can calculate the opposite sides length by the Pythagorean theorem. This means, if we use \$z\$ for the opposite side, that \$z^2+x^2=1^2=1\$. Solving for \$z\$ gives \$z=sqrt1-x^2\$.

Then \$sin(y)\$ is the ratio of the opposite size, \$z=sqrt1-x^2\$ & the hypotenuse, \$1\$. We may now say \$sin(arcos(x))=sqrt1-x^2\$.

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edited Apr 26, năm ngoái at 6:30
answered Apr 26, 2015 at 6:21

EoinEoin
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Draw a right triangle with one of the angles being \$cos^-1 x\$, then fill out the lengths of the sides of the triangle using the definition of cosine for two of the sides, & Pythagoras"s theorem for the third side. Then, use the definition of sine lớn find \$sin(cos^-1 x)\$. For questions lượt thích these, always try drawing a triangle first.

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answered Apr 26, 2015 at 6:15
Kevin HsuKevin Hsu
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\$\$y=sinarccos x\$\$\$\$x=cosarccos x\$\$

Therefore \$x\$ và \$y\$ are the sin and cos of the same angle and

\$\$x^2+y^2=1\$\$

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answered Apr 26, năm ngoái at 6:21
ajotatxeajotatxe
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