8. Consider the polar curve given by the equation \( r=2 \sin (\theta) \) a) (7pts) The point \( \left(\sqrt{2}, \frac{\pi}{4}\right) \) is a point on this polar curve. Determine the \( x \) )-coordinates of this point. b) (8pts) Represent this curve as a parametric curve (with parameter \( \theta \) ). c) (8pts) Set up, but do not evaluate, an integral which represents the area enclosed by this curve.

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (-7,0) \) and \( (7,0) \); vertices at \( (3,0) \) and \( (-3,0) \)

Write the complex number in trigonometric form \( r(\cos \theta+i \sin \theta) \), with \( \theta \) in the interval \( \left[0^{\circ}, 360^{\circ}\right) \). \[ \begin{array}{l}1+i\end{array} \] \( \begin{array}{l}1+i=\square\left(\cos \square^{\circ}+i \sin \square^{\circ}\right) \\ \text { (Type the value for } r \text { as an exact answer, using radicals as needed. Type the value for } \theta \text { as an integer or } \\ \text { as needed.) }\end{array} \)

Write the complex number in trigonometric form \( r(\cos \theta+i \sin \theta) \), with \( \theta \) in the interval \( \left[0^{\circ}, 360^{\circ}\right) \). \[ -3 \sqrt{3}+3 i \] \( -3 \sqrt{3}+3 i=\square\left(\cos \square^{\circ}+i \sin \square^{\circ}\right) \) (Type the value for \( r \) as an exact answer, using radicals as needed. Type the value for \( \theta \) as an intege as needed.)

Find the magnitude and positive direction angle of the vector \( \langle 2,-2 \sqrt{3}\rangle \). The magnitude of the vector is (Simplify your answer.)

Find the following product, and write the product in rectangular form, using exact values. \( \left[8\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]\left[3\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\right] \) \( \left[8\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]\left[3\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\right]=\square \) \( ( \) Type your answer in the form \( \mathrm{a}+\mathrm{bi} \).