91Ó°ÊÓ

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 1 \leq r \leq \sqrt{2 \sin (2 \theta)}, \frac{13 \pi}{12} \leq \theta \leq \frac{17 \pi}{12}\right\\} $$

In Exercises \(25-39\), find a parametric description for the given oriented curve. the ellipse \(9 x^{2}+4 y^{2}+24 y=0\), oriented clockwise (Shift the parameter so \(t=0\) corresponds to \((0,0) .)\)

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle 6,0\rangle $$

In Exercises \(25-39\), find a parametric description for the given oriented curve. the triangle with vertices \((0,0),(3,0),(0,4)\), oriented counter-clockwise (Shift the parameter so \(t=0\) corresponds to \((0,0) .)\)

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 0 \leq r \leq 2 \sqrt{3} \sin (\theta), 0 \leq \theta \leq \frac{\pi}{6}\right\\} \cup\left\\{(r, \theta) \mid 0 \leq r \leq 2 \cos (\theta), \frac{\pi}{6} \leq \theta \leq \frac{\pi}{2}\right\\} $$

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (7,-7) $$

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle-2.5,0\rangle $$

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=50 \operatorname{cis}\left(\pi-\arctan \left(\frac{7}{24}\right)\right) $$

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle 0, \sqrt{7}\rangle $$

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\frac{1}{2} \operatorname{cis}\left(\pi+\arctan \left(\frac{5}{12}\right)\right) $$