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Trigonometry : Complex Numbers

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Example Questions

Example Question #1 : Complex Numbers

Name the real part of this expression and the imaginary part of this expression: 3-7i .

Possible Answers:

Real:

Imaginary:

All parts are imaginary; there are no real parts

Real:

Imaginary: -7i

Real:

Imaginary:

All parts are real; there are no imaginary parts

Correct answer:

Real:

Imaginary: -7i

Explanation:

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 -7i .

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Example Question #1 : Complex Numbers

Find the product of the complex number and its conjugate:

4+2i

Possible Answers:

12+16i

4-2i

Correct answer:

Explanation:

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 4+2i is 4-2i . Now, we need to multiply these together using distribution, combining like terms, and substituting i^2=-1 .

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

$16-8i+8i-4i^{2}$

16+4

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Example Question #3 : Complex Numbers

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

Possible Answers:

-22i

$\sqrt{3}+22i$

$\sqrt{3-22i}$

$\sqrt{3}-22i$

$\sqrt{3+22i}$

Correct answer:

$\sqrt{3}+22i$

Explanation:

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 a+bi is a-bi . Therefore the conjugate of $\sqrt{3}-22i$ is $\sqrt{3}+22i$ .

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Example Question #1 : Complex Numbers

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

Possible Answers:

15+20i

28+16i

44i

14+30i

Correct answer:

14+30i

Explanation:

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

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

=(6+8)+(20i+10i)

=14+30i

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Example Question #2 : Complex Numbers

Simplify (6+5i)-(2-3i) .

Possible Answers:

4+8i

4+2i

8+8i

12i

Correct answer:

4+8i

Explanation:

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.

(6+5i)-(2-3i)

=(6-2)+(5i+3i)

=4+8i

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Example Question #5 : Complex Numbers

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

Possible Answers:

16+25i^2

16+40i+25i^2

9+40i

-9+40i

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 #6 : Complex Numbers

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

Possible Answers:

Complex conjugates do not exist for these terms

5; 3i

5; -3i

-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 #7 : Complex Numbers

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

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

Possible Answers:

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

-6+8i

9+6i

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

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 #8 : Complex Numbers

Which of the following represents 6-3i graphically?

Possible Answers:

Screen shot 2020 08 04 at 12.45.00 pm

Screen shot 2020 08 04 at 12.44.20 pm

Screen shot 2020 08 04 at 12.38.00 pm

Screen shot 2020 08 04 at 12.40.28 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 #2 : Complex Numbers

The following graph represents which one of the following?

Screen shot 2020 08 04 at 1.04.02 pm

Possible Answers:

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

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

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

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

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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Trigonometry : Complex Numbers

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Complex Numbers

Example Question #1 : Complex Numbers

Example Question #3 : Complex Numbers

Example Question #1 : Complex Numbers

Example Question #2 : Complex Numbers

Example Question #5 : Complex Numbers

Example Question #6 : Complex Numbers

Example Question #7 : Complex Numbers

Example Question #8 : Complex Numbers

Example Question #2 : Complex Numbers

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