91Ó°ÊÓ

Prove that the Law of Sines holds when \(\triangle A B C\) is a right triangle.

Convert the point from polar coordinates into rectangular coordinates. $$ (\pi, \arctan (\pi)) $$

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. $$ \vec{v}=\langle-\sqrt{2}, \sqrt{2}\rangle $$

In Exercises \(25-39\), find a parametric description for the given oriented curve. the circle \((x-3)^{2}+(y+1)^{2}=117\), oriented counter-clockwise

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 0 \leq r \leq 4 \cos (2 \theta),-\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}\right\\} $$

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=5 \operatorname{cis}\left(\arctan \left(\frac{4}{3}\right)\right) $$

Discuss with your classmates why knowing only the three angles of a triangle is not enough to determine any of the sides.

In Exercises \(25-39\), find a parametric description for the given oriented curve. the ellipse \((x-1)^{2}+9 y^{2}=9\), oriented counter-clockwise

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \left\\{(r, \theta) \mid 1 \leq r \leq 1-2 \cos (\theta), \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}\right\\} $$

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. $$ \vec{v}=\left\langle-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right\rangle $$