Activity 9.8.2.
(a)
For each of the points listed below, you should:
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Graph the point on a set of axes and label how each of the rectangular coordinates is measured.
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Draw and label how the polar coordinates are measured for each point.
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Compute the exact value of \(r\) and \(\theta\) for each point. Exact values include things like \(\sqrt{3}\text{,}\) \(\arcsin(3/4)\text{,}\) etc. Simplify trigonometric function values of any common angles you encounter, like \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\text{.}\)
(i)
\((x,y)=(-3,-7)\)
(ii)
\((x,y)=(7,-3)\)
(iii)
\((x,y)=(\sqrt{5},-2)\)
(iv)
\((x,y)=\left(\frac{3}{2},-\frac{3}{2}\right)\)
(v)
\((x,y)=\left(\frac{3}{2},\frac{3}{2}\right)\)
(b)
For each of the points listed below, you should:
-
Graph the point on a set of axes and label how each of the polar coordinates is measured.
-
Draw and label how the rectangular coordinates are measured for each point.
-
Compute the exact value of \(x\) and \(y\) for each point. Exact values include things like \(\sqrt{3}\text{,}\) \(\arcsin(3/4)\text{,}\) etc. Simplify trigonometric function values of any common angles you encounter, like \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\text{.}\)
(i)
\((r,\theta)=\left(1,-\frac{\pi}{6}\right)\)
(ii)
\((r,\theta)=\left(\sqrt{3},\arctan\left(\frac{\sqrt{2}}{2}\right)\right)\)
(iii)
\((r,\theta)=\left(-\sqrt{5},0\right)\)
(iv)
\((r,\theta)=\left(7,\frac{3\pi}{4}\right)\)
(v)
\((r,\theta)=\left(3,\pi-\frac{\pi}{6}\right)\)
