trigonometry and pre- calculus tutor worksheet 1 complex ... · numbers and combine like terms,...

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Trigonometry and Pre-

Calculus Tutor

Worksheet 1

Complex Numbers


Trigonometry and Pre-Calculus Tutor – Worksheet 1 – Complex Numbers

1. Add (5 + 8𝑖) + (9 − 6𝑖).

2. Add (−9 − 15𝑖) + (7 + 2𝑖).

3. Add 24 + (−13 − 17𝑖).

4. Add 16𝑖 + (−21 − 3𝑖).

5. Subtract (31 − 26𝑖) − (18 − 19𝑖).


6. Subtract (−14 + 9𝑖) − (−12 + 11𝑖).

7. Subtract −45𝑖 − (−24 − 17𝑖).

8. Subtract 41 − (−9 + 36𝑖).

9. Multiply (3 + 5𝑖)(7 + 8𝑖).

10. Multiply (6 − 4𝑖)(5 + 2𝑖).


11. Multiply (8 − 9𝑖)(5 − 2𝑖).

12. Multiply −5𝑖(8 + 𝑖).

13. What is the complex conjugate of 6 − 8𝑖?

14. What is the complex conjugate of 9 + 14𝑖?

15. Divide 9

6−3𝑖.


16. Divide 2

7+3𝑖.

17. Divide 14

28+42𝑖.

18. Divide 2−7𝑖

4+9𝑖.

19. Divide 5+𝑖

7−2𝑖.



9+12𝑖.


6+5𝑖.

22. What ordered pair in the coordinate plane corresponds to the graph of the

complex number 5 + 9𝑖?


complex number 3 − 8𝑖?


complex number −4 − 13𝑖?


25. Find the complex solutions to 𝑥2 + 5𝑥 + 11 = 0.

26. Find the complex solutions to 2𝑥2 − 7𝑥 + 12 = 0.

27. Find the complex solutions to 3𝑥2 + 11𝑥 + 13 = 0.


Answers – Trigonometry and Pre-Calculus Tutor – Worksheet 1 – Complex

Numbers

1. Add (5 + 8𝑖) + (9 − 6𝑖).

Separate the real component and the imaginary component in the complex

numbers and combine like terms, watching the sign of each term:

(5 + 9) + (8𝑖 − 6𝑖)

Answer: 14 + 2𝑖

2. Add (−9 − 15𝑖) + (7 + 2𝑖).



(−9 + 7) + (−15𝑖 + 2𝑖)

Answer: −2 − 13𝑖

3. Add 24 + (−13 − 17𝑖).



(24 − 13) + −17𝑖

Answer: 11 − 17𝑖

4. Add 16𝑖 + (−21 − 3𝑖).



−21 + (16𝑖 + −3𝑖)

Answer: −21 + 13𝑖


5. Subtract (31 − 26𝑖) − (18 − 19𝑖).



(31 − 18) + (−26𝑖 − −19𝑖)

(31 − 18) + (−26𝑖 + 19𝑖)


6. Subtract (−14 + 9𝑖) − (−12 + 11𝑖).



(−14 − −12) + (9𝑖 − 11𝑖)

(−14 + 12) + (9𝑖 − 11𝑖)

Answer: −2 − 2𝑖

7. Subtract −45𝑖 − (−24 − 17𝑖).



−(−24) + (−45𝑖 − −17𝑖)

24 + (−45𝑖 + 17𝑖)



8. Subtract 41 − (−9 + 36𝑖).



(41 − −9) − 36𝑖

(41 + 9) − 36𝑖


9. Multiply (3 + 5𝑖)(7 + 8𝑖).

Begin this problem by multiplying using the FOIL process. When multiplying

complex numbers, treat the 𝑖 as a variable unless it become 𝑖2. Because 𝑖2 = −1,

replace all 𝑖2 with −1.

(3 + 5𝑖)(7 + 8𝑖) = 3 ∙ 7 + 3 ∙ 8𝑖 + 5𝑖 ∙ 7 + 5𝑖 ∙ 8𝑖

= 21 + 24𝑖 + 35𝑖 + 40𝑖2

= 21 + 24𝑖 + 35𝑖 + 40 ∙ −1

= 21 + 24𝑖 + 35𝑖 − 40

Combine like terms.

= (21 − 40) + (24𝑖 + 35𝑖)

Answer: −19 + 59𝑖

10. Multiply (6 − 4𝑖)(5 + 2𝑖).




(6 − 4𝑖)(5 + 2𝑖) = 6 ∙ 5 + 6 ∙ 2𝑖 + −4𝑖 ∙ 5 + −4𝑖 ∙ 2𝑖


= 30 + 12𝑖 + −20𝑖 − 8𝑖2

= 30 + 12𝑖 + −20𝑖 − 8 ∙ −1

= 30 + 12𝑖 + −20𝑖 + 8

Combine like terms.

= (30 + 8) + (12𝑖 − 20𝑖)


11. Multiply (8 − 9𝑖)(5 − 2𝑖).




(8 − 9𝑖)(5 − 2𝑖) = 8 ∙ 5 + 8 ∙ −2𝑖 + −9𝑖 ∙ 5 + −9𝑖 ∙ −2𝑖

= 40 − 16𝑖 + −45𝑖 + 18𝑖2

= 40 − 16𝑖 + −45𝑖 + 18 ∙ −1

= 40 − 16𝑖 + −45𝑖 − 18

Combine like terms.

= (40 − 18) + (−16𝑖 − 45𝑖)


12. Multiply −5𝑖(8 + 𝑖).

Begin this problem by multiplying using distribution. When multiplying complex

numbers, treat the 𝑖 as a variable unless it become 𝑖2. Because 𝑖2 = −1, replace

all 𝑖2 with −1.

−5𝑖(8 + 𝑖) = −5𝑖 ∙ 8 + −5𝑖 ∙ 𝑖


= −40𝑖 − 5𝑖2

= −40𝑖 − 5 ∙ −1

= −40𝑖 + 5

Rearrange the result so it is in the form 𝑎 + 𝑏𝑖.


13. You are asked to find the complex conjugate of 6 − 8𝑖.

To find the complex conjugate of a complex number, simply change the sign

between the real component and the imaginary component. Warning: The

original complex number is NOT equal to its conjugate. The complex conjugate is:

6 + 8𝑖

14. You are asked to find the complex conjugate of 9 + 14𝑖.

To find the complex conjugate of a complex number, simply change the sign

between the real component and the imaginary component. Warning: The

original complex number is NOT equal to its conjugate. The complex conjugate is:

9 − 14𝑖

15. Divide 9

6−3𝑖.

The question asks you to divide, but the process really involves multiplying the

given fraction by an expression containing the complex conjugate of the

denominator in the numerator and in the denominator. Notice that complex


conjugates form a sum and a difference, so multiply them the same way as you

would multiply using a sum and a difference formula.

9

6 − 3𝑖=

9

6 − 3𝑖∙

6 + 3𝑖

6 + 3𝑖

9

6 − 3𝑖∙

6 + 3𝑖

6 + 3𝑖=

9 ∙ 6 + 9 ∙ 3𝑖

6 ∙ 6 − 3𝑖 ∙ 3𝑖

=54 + 27𝑖

36 − 9𝑖2=

54 + 27𝑖

36 − 9 ∙ −1

=54 + 27𝑖

36 + 9=

54 + 27𝑖

45

Simplify and convert to the form 𝑎 + 𝑏𝑖.

54 + 27𝑖

45=

6 + 3𝑖

5

Answer: 6

5+

3𝑖

5

16. Divide 2

7+3𝑖.





would multiply a sum and a difference.

2

7 + 3𝑖=

2

7 + 3𝑖∙

7 − 3𝑖

7 − 3𝑖

2

7 + 3𝑖∙

7 − 3𝑖

7 − 3𝑖=

2 ∙ 7 + 2 ∙ −3𝑖

7 ∙ 7 − 3𝑖 ∙ 3𝑖

=14 − 6𝑖

49 − 9𝑖2=

14 − 6𝑖

49 − 9 ∙ −1


=14 − 6𝑖

49 + 9=

14 − 6𝑖

58

Simplify and convert to the form 𝑎 + 𝑏𝑖.

14 − 6𝑖

58=

7 − 3𝑖

29

Answer: 7

29−

3𝑖

29

17. Divide 14

28+42𝑖.





would multiply a sum and a difference. However, the given fraction can be

simplified.

14

28 + 42𝑖=

1

2 + 3𝑖

1

2 + 3𝑖∙

2 − 3𝑖

2 − 3𝑖=

2 − 3𝑖

2 ∙ 2 − 3𝑖 ∙ 3𝑖

=2 − 3𝑖

4 − 9𝑖2=

2 − 3𝑖

4 − 9 ∙ −1=

2 − 3𝑖

4 + 9=

2 − 3𝑖

13

Convert to the form 𝑎 + 𝑏𝑖.

Answer: 2

13−

3𝑖

13



4+9𝑖.





would multiply a sum and a difference. Also, multiply the numerator using the

FOIL process.

2 − 7𝑖

4 + 9𝑖=

2 − 7𝑖

4 + 9𝑖∙

4 − 9𝑖

4 − 9𝑖

=2 ∙ 4 + 2 ∙ −9𝑖 + −7𝑖 ∙ 4 + −7𝑖 ∙ −9𝑖

4 ∙ 4 − 9𝑖 ∙ 9𝑖

=8 − 18𝑖 − 28𝑖 + 63𝑖2

16 − 81𝑖2

=8 − 18𝑖 − 28𝑖 + 63 ∙ −1

16 − 81 ∙ −1

=8 − 18𝑖 − 28𝑖 − 63

16 + 81

Combine like terms and simplify if possible.

=(8 − 63) + (−18𝑖 − 28𝑖)

97=

−55 − 46𝑖

97


Answer: −55

97−

46𝑖

97

19. Divide 5+𝑖

7−2𝑖.







FOIL process.

5 + 𝑖

7 − 2𝑖=

5 + 𝑖

7 − 2𝑖∙

7 + 2𝑖

7 + 2𝑖

=5 ∙ 7 + 5 ∙ 2𝑖 + 𝑖 ∙ 7 + 𝑖 ∙ 2𝑖

7 ∙ 7 − 2𝑖 ∙ 2𝑖

=35 + 10𝑖 + 7𝑖 + 2𝑖2

49 − 4𝑖2

=35 + 10𝑖 + 7𝑖 + 2 ∙ −1

49 − 4 ∙ −1

=35 + 10𝑖 + 7𝑖 − 2

49 + 4


=(35 − 2) + (10𝑖 + 7𝑖)

53=

33 + 17𝑖

53


Answer: 33

53+

17𝑖

53


9+12𝑖.






FOIL process. However, notice that the given expression can be simplified first.

9 − 12𝑖

9 + 12𝑖=

3 − 4𝑖

3 + 4𝑖


3 − 4𝑖

3 + 4𝑖=

3 − 4𝑖

3 + 4𝑖∙

3 − 4𝑖

3 − 4𝑖

=3 ∙ 3 + 3 ∙ −4𝑖 + −4𝑖 ∙ 3 + −4𝑖 ∙ −4𝑖

3 ∙ 3 − 4𝑖 ∙ 4𝑖

=9 − 12𝑖 − 12𝑖 + 16𝑖2

9 − 16𝑖2

=9 − 12𝑖 − 12𝑖 + 16 ∙ −1

9 − 16 ∙ −1

=9 − 12𝑖 − 12𝑖 − 16

9 + 16


=(9 − 16) + (−12𝑖 − 12𝑖)

25=

−7 − 24𝑖

25


Answer: −7

25−

24𝑖

25


6+5𝑖.






FOIL process.

7 − 3𝑖

6 + 5𝑖=

7 − 3𝑖

6 + 5𝑖∙

6 − 5𝑖

6 − 5𝑖

=7 ∙ 6 + 7 ∙ −5𝑖 + −3𝑖 ∙ 6 + −3𝑖 ∙ −5𝑖

6 ∙ 6 − 5𝑖 ∙ 5𝑖


=42 − 35𝑖 − 18𝑖 + 15𝑖2

36 − 25𝑖2

=42 − 35𝑖 − 18𝑖 + 15 ∙ −1

36 − 25 ∙ −1

=42 − 35𝑖 − 18𝑖 − 15

36 + 25


=(42 − 15) + (−35𝑖 − 18𝑖)

61=

27 − 53𝑖

61


Answer: 27

61−

53𝑖

61

22. The question asks you to identify the ordered pair in the coordinate plane

corresponds to the graph of the complex number 5 + 9𝑖.

The complex number 𝑎 + 𝑏𝑖 is graphed on the (𝑥, 𝑦) plane as (𝑎, 𝑏). Therefore,

the complex number 5 + 9𝑖 graphs as the point (5, 9).


corresponds to the graph of the complex number 3 − 8𝑖.


the complex number 3 − 8𝑖 graphs as the point (3, −8).


corresponds to the graph of the complex number −4 − 13𝑖.


the complex number −4 − 13𝑖 graphs as the point (−4, −13).


25. Find the complex solutions to 𝑥2 + 5𝑥 + 11 = 0.

A quadratic equation that does not intersect the 𝑥 −axis has no real solutions and

is generally not factorable. This type of quadratic equation has complex solutions

of the form 𝑎 + 𝑏𝑖 and is solved using the quadratic formula. Given the quadratic

equation of the form:

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

The quadratic formula is:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

In the given equation 𝑥2 + 5𝑥 + 11 = 0, 𝑎 = 1, 𝑏 = 5, and 𝑐 = 11. Plug these

values into the quadratic formula and solve for 𝑥.

𝑥 =−5 ± √(5)2 − 4 ∙ 1 ∙ 11

2 ∙ 1

𝑥 =−5 ± √25 − 44

2

𝑥 =−5 ± √−19

2=

−5 ± 𝑖√19

2

Express the solutions in the form 𝑎 + 𝑏𝑖.

Answer: 𝑥 = −5

2−

√19𝑖

2, 𝑥 = −

5

2+

√19𝑖

2







𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

In the given equation 2𝑥2 − 7𝑥 + 12 = 0, 𝑎 = 2, 𝑏 = −7, and 𝑐 = 12. Plug these


𝑥 =7 ± √(−7)2 − 4 ∙ 2 ∙ 12

2 ∙ 2

𝑥 =7 ± √49 − 96

4

𝑥 =7 ± √−47

4=

7 ± 𝑖√47

4


Answer: 𝑥 =7

4−

√47𝑖

4, 𝑥 =

7

4+

√47𝑖

4






𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎


In the given equation 3𝑥2 + 11𝑥 + 7 = 0, 𝑎 = 3, 𝑏 = 11, and 𝑐 = 13. Plug these


𝑥 =−11 ± √(11)2 − 4 ∙ 3 ∙ 13

2 ∙ 3

𝑥 =−11 ± √121 − 156

6

𝑥 =−11 ± √−35

6=

−11 ± 𝑖√35

6


Answer: 𝑥 = −11

6−

√35𝑖

6, 𝑥 = −

11

6+

√35𝑖

6






𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎



𝑥 =6 ± √(−6)2 − 4 ∙ 5 ∙ 8

2 ∙ 5


𝑥 =6 ± √36 − 160

10

𝑥 =6 ± √−124

10=

6 ± 2𝑖√31

10

Simplify.

6 ± 2𝑖√31

10=

3 ± 𝑖√31

5


Answer: 𝑥 =3

5−

√31𝑖

5, 𝑥 =

3

5+

√31𝑖

5






𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

In the given equation 6𝑥2 + 5𝑥 + 2 = 0, 𝑎 = 6, 𝑏 = 5, and 𝑐 = 2. Plug these


𝑥 =−5 ± √(5)2 − 4 ∙ 6 ∙ 2

2 ∙ 6


𝑥 =−5 ± √25 − 48

12

𝑥 =−5 ± √−23

12=

−5 ± 𝑖√23

12


Answer: 𝑥 = −5

12−

√23𝑖

12, 𝑥 = −

5

12+

√23𝑖

12






𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎



𝑥 =8 ± √(−8)2 − 4 ∙ 3 ∙ 10

2 ∙ 3

𝑥 =8 ± √64 − 120

6

𝑥 =8 ± √−56

6=

8 ± 2𝑖√14

6

Simplify.


8 ± 2𝑖√14

6=

4 ± 𝑖√14

3


Answer: 𝑥 =4

3−

√14𝑖

3, 𝑥 =

4

3+

√14𝑖

3

trigonometry and pre- calculus tutor worksheet 1 complex ... · numbers and combine like terms,...

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