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Trigonometry : Complex Numbers/Polar Form

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Example Question #6 : Complex Numbers

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

Possible Answers:

9+40i

-9+40i

16+40i+25i^2

16+25i^2

Correct answer:

-9+40i

Explanation:

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.)

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

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

$16+20i+20i+25i^{2}$

16+40i-25 (recall that i^2=-1 )

-9+40i

Please note that while the answer choice 16+40i+25i^2 is not incorrect, it is not fully simplified and therefore not the correct choice.

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Example Question #7 : Complex Numbers

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

Possible Answers:

5; -3i

5; 3i

Complex conjugates do not exist for these terms

-5; -3i

-5; 3i

Correct answer:

5; -3i

Explanation:

While these terms may not look like they follow the typical format of a+bi , don't let them fool you! We can read 5 as 5+0i and we can read 3i as 0+3i . Now recalling that the complex conjugate of a+bi is a-bi , we can see that the complex conjugate of is just and the complex conjugate of is -3i

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Example Question #8 : Complex Numbers

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

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

Possible Answers:

-6+8i

9+6i

$\frac{13+12i}{39}$

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

Correct answer:

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

Explanation:

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:

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

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

$=\frac{27+18i+18i+12i^{2}}{81+54i-54i-36i^{2}}$

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

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

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

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

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Example Question #9 : Complex Numbers

Which of the following represents 6-3i graphically?

Possible Answers:

Screen shot 2020 08 04 at 12.40.28 pm

Screen shot 2020 08 04 at 12.45.00 pm

Screen shot 2020 08 04 at 12.38.00 pm

Screen shot 2020 08 04 at 12.44.20 pm

Correct answer:

Screen shot 2020 08 04 at 12.38.00 pm

Explanation:

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 6-3i , 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 6-3i , but we can also represent it by drawing a vector from the origin to point P. Both representations are in the diagram below.

Screen shot 2020 08 04 at 12.38.00 pm

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Example Question #10 : Complex Numbers

The following graph represents which one of the following?

Screen shot 2020 08 04 at 1.04.02 pm

Possible Answers:

$\left (4+2i \right )+\left ( 1+3i \right )=\left ( 6+4i \right )$

$\left (4+3i \right )+\left ( 1+2i \right )=\left ( 5+5i \right )$

$\left (4+2i \right )+\left ( 3+i \right )=\left ( 7+3i \right )$

$\left (4+i \right )+\left ( 2+3i \right )=\left ( 6+4i \right )$

Correct answer:

$\left (4+i \right )+\left ( 2+3i \right )=\left ( 6+4i \right )$

Explanation:

We can take any complex number a+bi and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore V_1=4+i . Likewise, V_2=2+3i . Then, we can add these two vectors together, summing their real parts and their imaginary parts to create their resultant vector R=6+4i . Therefore the correct answer is $\left (4+i \right )+\left ( 2+3i \right )=\left ( 6+4i \right )$ .

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Example Question #11 : Complex Numbers

The following graph represents which one of the following?

Screen shot 2020 08 04 at 1.05.05 pm

Possible Answers:

V_2-V_1=R

V_1+V_2=R

$V_1\cdot V_2=R$

V_1-V_2=R

Correct answer:

V_1-V_2=R

Explanation:

The above image is a graphic representation of subtraction of complex numbers (which are represented by vectors V_1 and V_2 . We can take any complex number a+bi and graph it as a vector, measuring units in the x direction and units in the y direction. Therefore V_1=4+i . Likewise, V_2=2+3i .

To help us visualize subtraction, instead of thinking about taking V_1-V_2 , we should instead visualize V_1+(-V_2) . The below figure shows -V_2 with a dotted line. Visually, the resultant vector R=V_1+(-V_2) lies in between the vectors V_1 and -V_2 . Algebraically, we get V_1-V_2 =(4+i)-(2+3i)=2-2i or =(4+i)+(-2-3i)=2-2i . Either way you think about it, the resulting vector is R=2-2i

Screen shot 2020 08 04 at 1.05.55 pm

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Trigonometry : Complex Numbers/Polar Form

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #6 : Complex Numbers

Example Question #7 : Complex Numbers

Example Question #8 : Complex Numbers

Example Question #9 : Complex Numbers

Example Question #10 : Complex Numbers

Example Question #11 : Complex Numbers

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