# Inverse Trig Functions Integration Homework

Since $\ds \sin^2y \cos^2 y=1$, $\ds \cos^2y=1-\sin^2y=1-x^2$.

Since $\ds \sin^2y \cos^2 y=1$, $\ds \cos^2y=1-\sin^2y=1-x^2$.So $\ds \cos y=\pm\sqrt$, but which is it—plus or minus?It is not true that the arcsine undoes the sine, for example, $\sin(5\pi/6)=1/2$ and $\arcsin(1/2)=\pi/6$, so doing first the sine then the arcsine does not get us back where we started.

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Recall that a function and its inverse undo each other in either order, for example, $\ds (\root3\of x)^3=x$ and $\ds \root3\of=x$.Consider the functions y cosx and y cos 1 o a Fill in the table below for the cosine function on the restricted interval 0 x r. In addition, a set of answer pages no shown work, just the answer comes with the solution manual for the homework problems. Learn about the definition of the basic trigonometric functions sinx, cosx, and tanx, and use advanced trigonometric functions for various purposes. Finally we look at the tangent; the other trigonometric functions also have "partial inverses'' but the sine, cosine and tangent are enough for most purposes.The tangent, truncated tangent and inverse tangent are shown in figure 4.9.3; the derivative of the arctangent is left as an exercise.It could in general be either, but this isn't "in general'': since $y=\arcsin(x)$ we know that $-\pi/2\le y\le \pi/2$, and the cosine of an angle in this interval is always positive.Thus $\ds \cos y=\sqrt$ and $$\arcsin(x)=.$$ Note that this agrees with figure 4.9.1: the graph of the arcsine has positive slope everywhere. As with the sine, we must first truncate the cosine so that it can be inverted, as shown in figure 4.9.2.Then we use implicit differentiation to find that $$\arccos(x)=.$$ Note that the truncated cosine uses a different interval than the truncated sine, so that if $y=\arccos(x)$ we know that [[

Recall that a function and its inverse undo each other in either order, for example, $\ds (\root3\of x)^3=x$ and $\ds \root3\of=x$.

Consider the functions y cosx and y cos 1 o a Fill in the table below for the cosine function on the restricted interval 0 x r. In addition, a set of answer pages no shown work, just the answer comes with the solution manual for the homework problems.

Learn about the definition of the basic trigonometric functions sinx, cosx, and tanx, and use advanced trigonometric functions for various purposes.

Finally we look at the tangent; the other trigonometric functions also have "partial inverses'' but the sine, cosine and tangent are enough for most purposes.

The tangent, truncated tangent and inverse tangent are shown in figure 4.9.3; the derivative of the arctangent is left as an exercise.

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Recall that a function and its inverse undo each other in either order, for example, $\ds (\root3\of x)^3=x$ and $\ds \root3\of=x$.Consider the functions y cosx and y cos 1 o a Fill in the table below for the cosine function on the restricted interval 0 x r. In addition, a set of answer pages no shown work, just the answer comes with the solution manual for the homework problems. Learn about the definition of the basic trigonometric functions sinx, cosx, and tanx, and use advanced trigonometric functions for various purposes. Finally we look at the tangent; the other trigonometric functions also have "partial inverses'' but the sine, cosine and tangent are enough for most purposes.The tangent, truncated tangent and inverse tangent are shown in figure 4.9.3; the derivative of the arctangent is left as an exercise.It could in general be either, but this isn't "in general'': since $y=\arcsin(x)$ we know that $-\pi/2\le y\le \pi/2$, and the cosine of an angle in this interval is always positive.Thus $\ds \cos y=\sqrt$ and $$\arcsin(x)=.$$ Note that this agrees with figure 4.9.1: the graph of the arcsine has positive slope everywhere. As with the sine, we must first truncate the cosine so that it can be inverted, as shown in figure 4.9.2.Then we use implicit differentiation to find that $$\arccos(x)=.$$ Note that the truncated cosine uses a different interval than the truncated sine, so that if $y=\arccos(x)$ we know that $0\le y\le \pi$.The computation of the derivative of the arccosine is left as an exercise. When given integral problems, look for these patterns. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

]]\le y\le \pi\$.The computation of the derivative of the arccosine is left as an exercise. When given integral problems, look for these patterns. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

## Comments Inverse Trig Functions Integration Homework

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