Polar Coordinates and Complex Numbers

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Pre-Calculus › Polar Coordinates and Complex Numbers

Questions 1 - 10

1

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 $(-5, -5\sqrt3)$

2

Evaluate:

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

Explanation

To evaluate this problem we need to FOIL the binomials.

Now recall that

Thus,

3

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

4

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

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

.

5

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.

6

Write the equation in polar form

$r = 2+3 \sin \theta$

$r = 3 \sin \theta$

$r = 4 + 3 \sin \theta$

$r = 4 - 3 \sin \theta$

$r = 2 - 3 \sin \theta$

Explanation

First re-arrange the original equation so that the 4 is factored out on the right side, and put and next to each other:

Make the substitutions and $y = r \sin \theta$ :

$(r^2 - 3 r \sin \theta )^2 = 4r^2$ take the square root of both sides

$r^2 - 3 r \sin \theta = 2 r$ divide both sides by r

$r - 3 \cos \theta = 2$ add $3 \cos \theta$ to both sides

$r = 2 + 3 \cos \theta$

7

Convert from polar form to rectangular form:

$r=4\cos\theta$

Explanation

Start by multiplying both sides by .

$r=4\cos\theta$

$r^2=4r\cos\theta$

Keep in mind that

$x^2+y^2=4r\cos\theta$

Remember that $x=r\cos\theta$

So then,

Now, complete the square.

8

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

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

.

9

Evaluate:

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

Explanation

To evaluate this problem we need to FOIL the binomials.

Now recall that

Thus,

10

Evaluate:

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

Explanation

To evaluate this problem we need to FOIL the binomials.

Now recall that

Thus,

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