Trigonometric and Polar Functions — Quiz 2

Questions in this quiz (7)

1. Which of the following choices is equivalent to the complex number $$3\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$$?

   • $$\frac{3}{2}+\frac{3\sqrt{3}}{2}i$$
   • $$\frac{3\sqrt{3}}{2}+\frac{3}{2}i$$
   • $$\sqrt{3}+3i$$
   • $$3+\sqrt{3}i$$

2. Which of the following choices represents the complex number $$k+ki$$, where $$k>0$$, in polar form?

   • $$k\sqrt{2}\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)$$
   • $$-k\sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$$
   • $$k\sqrt{2}\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$$
   • $$-k\sqrt{2}\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)$$

3. Which of the following choices is an equivalent polar form of the complex number $$1-i\sqrt{3}$$?

   • $$2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$$
   • $$2\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$$
   • $$2\left(\cos\left(-\frac{\pi}{3}\right)+i\sin\left(-\frac{\pi}{3}\right)\right)$$
   • $$2\left(\cos\left(-\frac{2\pi}{3}\right)+i\sin\left(-\frac{2\pi}{3}\right)\right)$$

4. Write $$-3i$$ in polar form.

   • $$3\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)$$
   • $$3\left(\cos\pi+i\sin\pi\right)$$
   • $$3\left(\cos 0+i\sin 0\right)$$
   • $$3\left(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}\right)$$

5. Which of the following choices is the equivalent Cartesian form of the polar form $$2\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)$$?

   • $$-1+\sqrt{3}i$$
   • $$1-\sqrt{3}i$$
   • $$-1-\sqrt{3}i$$
   • $$1+\sqrt{3}i$$

6. Which of the following complex numbers cannot be written in polar form?

   • $$i$$
   • $$2-3i$$
   • $$0$$
   • $$-4$$

7. Find the average rate of change for the polar curve $$r=3+2\cos\theta$$ on the interval $$[0,\pi]$$.

   • $$\frac{4}{\pi}$$
   • $$\frac{2}{\pi}$$
   • $$-\frac{4}{\pi}$$
   • $$-\frac{2}{\pi}$$