Complex Numbers/Polar Form - Trigonometry

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Question

Name the real part of this expression and the imaginary part of this expression: .

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Answer

The real part of this expression includes any terms that do not have attached to them. Therefore the real part of this expression is 3. The imaginary part of this expression includes any terms with that cannot be further reduced; the imaginary part of this expression is .

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Question

Find the product of the complex number and its conjugate:

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Answer

To solve this problem, we must first identify the conjugate of this complex number. The conjugate keeps the real portion of the number the same, but changes the sign of the imaginary part of the number. Therefore the conjugate of is . Now, we need to multiply these together using distribution, combining like terms, and substituting .

$\left ( 4+2i \right )\left ( 4-2i \right )$

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Question

What is the complex conjugate of $$\sqrt{3}$-22i$ ?

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Answer

To solve this problem, we must understand what a complex conjugate is and how it relates to a complex number. The conjugate of a number is . Therefore the conjugate of $$\sqrt{3}$-22i$ is $$\sqrt{3}$+22i$ .

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Question

Simplify $\left ( 6+20i \right )+\left ( 8+10i \right )$ .

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Answer

To add complex numbers, we must combine like terms: real with real, and imaginary with imaginary.

$\left ( 6+20i \right )+\left ( 8+10i \right )$

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Question

Simplify .

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Answer

In order to solve this problem, we must combine real numbers with real numbers and imaginary numbers with imaginary numbers. Be careful to distribute the subtraction sign to all terms in the second set of parentheses.

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Question

Simplify $\left ( 4+5i \right $)^{2}$$ .

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Answer

To solve this problem, make sure you set it up to multiply the entire parentheses by itself (a common mistake it to try to simply distribute the exponent 2 to each of the terms in the parentheses.)

$\left ( 4+5i \right $)^{2}$$

$\left ( 4+5i \right )\left ( 4+5i \right )$

(recall that )

Please note that while the answer choice is not incorrect, it is not fully simplified and therefore not the correct choice.

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Question

What is the complex conjugate of 5? What is the complex conjugate of 3i?

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Answer

While these terms may not look like they follow the typical format of , don't let them fool you! We can read 5 as and we can read 3i as . Now recalling that the complex conjugate of is , we can see that the complex conjugate of is just and the complex conjugate of is

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Question

Perform division on the following expression by utilizing a complex conjugate:

$\frac{3+2i}{9-6i}$

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Answer

To perform division on complex numbers, multiple both the numerator and the denominator of the fraction by the complex conjugate of the denominator. This looks like:

$\frac{3+2i}{9-6i}$\cdot $\frac{9+6i}{9+6i}$

$=\frac{\left ( 3+2i \right )\left ( 9+6i \right )}{\left ( 9-6i \right )\left ( 9+6i \right )}$$

$=\frac{27+36i-12}{81+36}$$

$=\frac{15+36i}{117}$$

$=\frac{3\left( 5+12i \right)}{3\left( 39 \right)}$$

$=\frac{5+12i}{39}$$

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Question

Which of the following represents graphically?

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Answer

To represent complex numbers graphically, we treat the x-axis as the "axis of reals" and the y-axis as the "axis of imaginaries." To plot , we want to move 6 units on the x-axis and -3 units on the y-axis. We can plot the point P to represent , but we can also represent it by drawing a vector from the origin to point P. Both representations are in the diagram below.

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Question

The following graph represents which one of the following?

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Answer

We can take any complex number and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore . Likewise, . Then, we can add these two vectors together, summing their real parts and their imaginary parts to create their resultant vector . Therefore the correct answer is $\left (4+i \right )+\left ( 2+3i \right )=\left ( 6+4i \right )$ .

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Question

The following graph represents which one of the following?

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Answer

The above image is a graphic representation of subtraction of complex numbers (which are represented by vectors and . We can take any complex number and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore . Likewise, .

To help us visualize subtraction, instead of thinking about taking , we should instead visualize . The below figure shows with a dotted line. Visually, the resultant vector lies in between the vectors and . Algebraically, we get or . Either way you think about it, the resulting vector is

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Question

The polar coordinates $\left ( r,\theta \right )$ of a point are $\left(2.1, $\frac{7\pi }{3}\right )$ . Convert these polar coordinates to rectangular coordinates.

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Answer

Given the polar coordinates $(r,\theta )$ , the -coordinate is $x= r \cos \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = $\frac{7\pi }{3}$$ :

$x = r \cos \theta = 2.1 \cdot \cos $\frac{7\pi }{3}$ = 2.1 \cdot 0.5 = 1.05$

Likewise, given the polar coordinates $(r,\theta )$ , the -coordinate is $y= r \sin \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = $\frac{7\pi }{3}$$ :

$y = r \sin \theta = 2.1 \cdot \sin $\frac{7\pi }{3}$ \approx 2.1 \cdot 0.8660 \approx 1.82$

Therefore the rectangular coordinates of the point $\left(2.1, $\frac{7\pi }{3}\right )$ are .

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Question

Express the complex number $z=4(\cos240^\circ+i\sin240^\circ)$ in rectangular form.

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Answer

To convert this number to rectangular form, first think about what $\cos240^\circ$ and $\sin240^\circ$ are equal to. Because $240^\circ=180^\circ+60^\circ$ , we can use a 30-60-90o reference triangle in the 3rd quadrant to determine these values.

$\cos240^\circ=-$\frac{1}{2}$$

$\sin240^\circ=-$\frac{\sqrt{3}$}{2}$

Now plug these in and continue solving:

$z=4(\cos240^\circ+i\sin240^\circ)$

$z=4(-$\frac{1}{2}$-i$\frac{\sqrt{3}$}{2})$

$z=-2-2$\sqrt{3}$i$

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Question

For the complex number , find the modulus and the angle $\theta=\tan^$-^1$\left ($\frac{y}{x}$ \right )$ . Then, express this number in polar form $z=r(\cos\theta+i\sin\theta)$ .

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Answer

This problem has given us formulas, so we just need to plug in and and solve.

$r=$\sqrt{218}$$

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

$\theta=\tan^$-^1$\left ($\frac{13}{7}$ \right )$

$\theta=61.7^\circ$

$z=r(\cos\theta+i\sin\theta)$

$z=$\sqrt{218}$(\cos61.7^\circ+i\sin61.7^\circ)$

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Question

Express the complex number $z=3(\cos30^\circ+i\sin30^\circ)$ in rectangular form .

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Answer

To convert this number to rectangular form, first think about what $\cos30^\circ$ and $\sin30^\circ$ are equal to. We can use a 30-60-90o reference triangle in the 1st quadrant to determine these values.

$\cos30^\circ=\frac{\sqrt{3}$}{2}$

$\sin30^\circ=\frac{1}{2}$$

Next, plug these values in and simplify:

$z=3(\cos30^\circ+i\sin30^\circ)$

$z=3($\frac{\sqrt{3}$}{2}+\frac{1}{2}$i)$

$z=\frac{3\sqrt{3}$}{2}+\frac{3}{2}$i$

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Question

For the complex number , find the modulus and the angle $\theta=\tan^$-^1$\left ($\frac{y}{x}$ \right )$ . Then, express this number in polar form $z=r(\cos\theta+i\sin\theta)$ .

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Answer

This problem has given us formulas, so we just need to plug in and and solve.

$r=$\sqrt{29}$$

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

$\theta=\tan^$-^1$\left ($\frac{-5}{2}$ \right )$

$\theta=-68.2^\circ$

$z=r(\cos\theta+i\sin\theta)$

$z=$\sqrt{29}$(\cos(-68.2^\circ)+i\sin(-68.2^\circ))$

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Question

Express the complex number in polar form.

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Answer

In order to complete this problem, you must understand three formulas that allow you to convert from the rectangular form of a complex number to the polar form of a complex number. These formulas are , $\theta=\tan^$-^1$\left ($\frac{y}{x}$ \right )$ , and the polar form $z=r(\cos\theta+i\sin\theta)$ . Additionally, understand that based on the given info, and . Begin by finding the modulus:

$r=$\sqrt{17}$$

Next, let's find the angle $\theta$ , also referred to as the amplitude of the complex number.

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

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

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

$\theta=14.04^\circ$

Finally, plug each of these into the polar form of a complex number:

$z=r(\cos\theta+i\sin\theta)$

$z=$\sqrt{17}$(\cos14.04^\circ+i\sin14.04^\circ)$

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Question

Multiply the following complex numbers (in polar form), giving the result in both polar and rectangular form.

$2(\cos70^\circ+i\sin70^\circ)\cdot 3(\cos20^\circ+i\sin20^\circ)$

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Answer

The modulus of the product of two complex numbers is the product of their moduli, and the amplitude of the product is the sum of their amplitudes.

Therefore, the new modulus will be $2\cdot 3=6$ and the new amplitude will be $70^\circ+20^\circ=90^\circ$ . Therefore

$2(\cos70^\circ+i\sin70^\circ)\cdot 3(\cos20^\circ+i\sin20^\circ)$ $=6(\cos90^\circ+i\sin90^\circ)$

We must also express this in rectangular form, which we can do by substituting $\cos90^\circ=0$ and $\sin90^\circ=1$ . We get:

$6(\cos90^\circ+i\sin90^\circ)$

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Question

Find the following quotients, given that $z_1=10(\cos40^\circ+i\sin40^\circ)$ and $z_2=2(\cos220^\circ-i\sin220^\circ)$ . Give results in both polar and rectangular forms.

(a) $\frac{z_1}{z_2}$

(b) $\frac{z_2}{z_1}$

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Answer

The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modulus of the divisor. The amplitude of the quotient is the amplitude of the dividend minus the amplitude of hte divisor.

(a) The modulus for $\frac{z_1}{z_2}$ is equal to $$\frac{10}{2}$=5$ . The amplitude for $\frac{z_1}{z_2}$ is equal to $40^\circ-220^\circ=-180^\circ=180^\circ$ . (We have chosen to represent this as the coterminal angle $180^\circ$ rather than $-180^\circ$ as it is more conventional to represent angle measures as a positive angle between $0^\circ$ and $360^\circ$ .) Putting this together, we get $$\frac{z_1}{z_2}$=5(\cos180^\circ+i\sin180^\circ)$ . To represent this in rectangular form, substitute $\cos180^\circ=-1$ and $\sin180^\circ=0$ to get $$\frac{z_1}{z_2}$=5(-1+0i)=-5$ .

(b) The modulus for $\frac{z_2}{z_1}$ is equal to $\frac{2}{10}$=\frac{1}{5}$ . The amplitude for $\frac{z_2}{z_1}$ is equal to $220^\circ-40^\circ=180^\circ$ . Putting this together, we get $$\frac{z_2}{z_1}$=\frac{1}{5}$(\cos180^\circ+i\sin180^\circ)$ . To represent this in rectangular form, substitute $\cos180^\circ=-1$ and $\sin180^\circ=0$ to get $\frac{z_2}{z_1}$=\frac{1}{5}$(-1+0i)=-$\frac{1}{5}$ .

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Question

The polar coordinates $\left ( r,\theta \right )$ of a point are $\left(2.1, $\frac{7\pi }{3}\right )$ . Convert these polar coordinates to rectangular coordinates.

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Answer

Given the polar coordinates $(r,\theta )$ , the -coordinate is $x= r \cos \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = $\frac{7\pi }{3}$$ :

$x = r \cos \theta = 2.1 \cdot \cos $\frac{7\pi }{3}$ = 2.1 \cdot 0.5 = 1.05$

Likewise, given the polar coordinates $(r,\theta )$ , the -coordinate is $y= r \sin \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = $\frac{7\pi }{3}$$ :

$y = r \sin \theta = 2.1 \cdot \sin $\frac{7\pi }{3}$ \approx 2.1 \cdot 0.8660 \approx 1.82$

Therefore the rectangular coordinates of the point $\left(2.1, $\frac{7\pi }{3}\right )$ are .

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