Polar Coordinates and Complex Numbers

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Pre-Calculus › Polar Coordinates and Complex Numbers

Questions 1 - 10

1

Convert to polar coordinates.

$\left(5,\frac{\pi}{2}\right)$

$(\sqrt5,0)$

$(\sqrt5,\pi)$

Explanation

Write the Cartesian to polar conversion formulas.

$\theta=tan^-^ \left(\frac{y}{x}\right)$

Substitute the coordinate point to the equations to solve for $(r,\theta)$ .

$r=\sqrt{5^2+0^2}=5$

$\theta=tan^-^1\left(\frac{y}{x}\right)=tan^-^1 \left(\frac{0}{5}\right)=0$

Ensuring that is located the first quadrant, the correct angle is zero.

Therefore, the answer is .

2

How could you express $\left(10, \frac{\pi}{10}\right)$ in rectangular coordinates?

Round to the nearest hundredth.

Explanation

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

Polar to rectangular a

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

$cos\left(\frac{\pi}{10}\right) = \frac{x}{10}$

$0.951 \approx \frac{x}{10}$ multiply both sides by 10.

We can figure out the y-coordinate by using the sine:

$sin\left(\frac{\pi}{10}\right) = \frac{y}{10}$

$0.309 \approx \frac{y}{10}$

3

Convert the polar coordinates to rectangular form:

$\left(10, \frac{4\pi}{3}\right)$

$(-5, -5\sqrt3)$

$(-5\sqrt3, -5)$

$(5\sqrt3, -5)$

$(5, 5\sqrt3)$

Explanation

To convert polar coordinates $(r, \theta)$ to rectangular coordinates ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=10\cos \frac{4\pi}{3}=10\left(-\frac{1}{2}\right)=-5$

$y=10 \sin \frac{4\pi}{3}=10\left(-\frac{\sqrt3}{2}\right)=-5\sqrt3$

The rectangular coordinates are $(-5, -5\sqrt3)$

4

Convert to polar coordinates.

$\left(5,\frac{\pi}{2}\right)$

$(\sqrt5,0)$

$(\sqrt5,\pi)$

Explanation

Write the Cartesian to polar conversion formulas.

$\theta=tan^-^ \left(\frac{y}{x}\right)$

Substitute the coordinate point to the equations to solve for $(r,\theta)$ .

$r=\sqrt{5^2+0^2}=5$

$\theta=tan^-^1\left(\frac{y}{x}\right)=tan^-^1 \left(\frac{0}{5}\right)=0$

Ensuring that is located the first quadrant, the correct angle is zero.

Therefore, the answer is .

5

Evaluate:

$(2+i)\cdot(1+4i)$

Explanation

To evaluate this problem we need to FOIL the binomials.

Now recall that

Thus,

6

Evaluate:

$(2+i)\cdot(1+4i)$

Explanation

To evaluate this problem we need to FOIL the binomials.

Now recall that

Thus,

7

Convert to polar form.

$\left(3\sqrt2,\frac{5\pi}{4}\right)$

$\left(3\sqrt2,\frac{\pi}{4}\right)$

$\left(3,\frac{\pi}{4}\right)$

$\left(3,\frac{5\pi}{4}\right)$

$\left(-3,-\frac{\pi}{4}\right)$

Explanation

Write the Cartesian to polar conversion formulas.

$\theta=tan^-^1 \left(\frac{y}{x}\right)$

Substitute the coordinate point to the equations to find $(r,\theta)$ .

$r=\sqrt{(-3)^2+(-3)^2}=\sqrt{18}=3\sqrt2$

$\theta=tan^-^1 \left(\frac{y}{x}\right)=tan^-^1 \left(\frac{-3}{-3}\right)=\frac{\pi}{4}$

Since is not located in between the first quadrant, this is not the correct angle. The correct location of this coordinate is in the third quadrant. Add $\pi$ radians to get the correct angle.

$\theta=\frac{\pi}{4}+\pi=\frac{5\pi}{4}$

Therefore, the answer is $\left(3\sqrt2,\frac{5\pi}{4}\right)$ .

8

Convert the polar coordinates to rectangular coordinates:

$\left(5, \frac{2\pi}{3}\right)$

$\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

$\left(\frac{2}{5}, \frac{\sqrt3}{2}\right)$

$\left(\frac{\sqrt2}{4}, \frac{5}{\sqrt3}\right)$

Explanation

To convert polar coordinates $(r, \theta)$ to rectangular coordinates ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=5\cos \frac{2\pi}{3}=5\left(-\frac{1}{2}\right)=-\frac{5}{2}$

$y=5 \sin \frac{2\pi}{3}=5\left(\frac{\sqrt3}{2}\right)=\frac{5\sqrt3}{2}$

The rectangular coordinates are $\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

9

What is the magnitude of ?

$\sqrt13$

Explanation

To find the magnitude of a complex number we use the following formula:

$|z|=\sqrt{a^2+b^2}$ , where .

Therefore we get,

$|z|=\sqrt{2^2+3^2}=\sqrt{13}$ .

Now to find

$|z|^2=(\sqrt{13})^2$

.

10

Find the magnitude of :

${}(1+3i)^{7}$ , where the complex number satisfies ${}i^{2}=-1$ .

${}10^3\sqrt{10}$

${}10^7\sqrt{10}$

${}10\sqrt{10}$

${}10^{14}\sqrt{10}$

${}10^2\sqrt{10}$

Explanation

Note for any complex number z, we have:

${}|z^n|=|z|^n$ .

Let . Hence ${}|z|=\sqrt{10}$

Therefore:

${}|z^{7}|=|z|^{7}=(\sqrt{10})^{7}=(\sqrt{10})^{6}(\sqrt{10})$

This gives the result.

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