1. Trigonometric Form Section 6.5 Precalculus PreAP/Dual, Revised ©2016
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§6.5: Trig Form of Complex Number

2. §6.5: Trig Form of Complex Number
Review
Solve 𝒚= 𝒙 𝟐 −𝟏 and graph on a coordinate plane
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§6.5: Trig Form of Complex Number

3. §6.5: Trig Form of Complex Number
Review
Solve 𝒚= 𝒙 𝟐 +𝟏 and graph on a coordinate plane
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§6.5: Trig Form of Complex Number

4. §6.5: Trig Form of Complex Number
The Complex Plane
Graphing the Complex Plane
Imaginary Axis
Real Axis O
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§6.5: Trig Form of Complex Number

5. §6.5: Trig Form of Complex Number
Example 1
Graph each complex number: 𝟐−𝟑𝒊 •
2 – 3i
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§6.5: Trig Form of Complex Number

6. §6.5: Trig Form of Complex Number
Example 2
Graph each complex number: 𝟐−𝒊 𝟐+𝒊 • 5
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§6.5: Trig Form of Complex Number

7. §6.5: Trig Form of Complex Number
Your Turn
Graph each complex number: 𝟐𝒊 𝟐−𝒊 •
2 + 4i
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§6.5: Trig Form of Complex Number

8. Absolute Value of Complex Numbers
Measures the distance from the origin of the point and into the complex plane, as listed as 𝒂+𝒃𝒊 or 𝒙+𝒚𝒊
Equation: 𝒙+𝒚𝒊 = 𝒙 𝟐 + 𝒚 𝟐
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§6.5: Trig Form of Complex Number

9. §6.5: Trig Form of Complex Number
The Complex Plane
Graphing the Complex Plane
Imaginary Axis
Real Axis O
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§6.5: Trig Form of Complex Number

10. §6.5: Trig Form of Complex Number
Example 3
Solve |𝟑+𝟒𝒊|
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§6.5: Trig Form of Complex Number

11. §6.5: Trig Form of Complex Number
Example 4
Solve −𝟐+𝟒𝒊
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§6.5: Trig Form of Complex Number

12. §6.5: Trig Form of Complex Number
Your Turn
Solve |𝟏𝟎−𝟒𝒊|
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§6.5: Trig Form of Complex Number

13. §6.5: Trig Form of Complex Number
Trigonometric Form
Identify the relationship of 𝒙, 𝒚, 𝒓, and 𝜽 in terms of trig functions
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§6.5: Trig Form of Complex Number

14. §6.5: Trig Form of Complex Number
Trigonometric Form
Identify the relationship of 𝒙, 𝒚, 𝒓, and 𝜽 in terms of trig functions
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§6.5: Trig Form of Complex Number

15. §6.5: Trig Form of Complex Number
Trigonometric Form
Polar Form or “Trigonometric Form” the equation is 𝒓 𝐜𝐨𝐬 𝜽 +𝒊 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 +𝒊 𝐬𝐢𝐧 𝜽
Short-hand written as 𝒓 𝐜𝐢𝐬 𝜽
The number 𝒓 is called the modulus or absolute value of the complex number
Angle 𝜽 is called the argument of the complex number
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§6.5: Trig Form of Complex Number

16. Steps in Converting in Trigonometric Form
Establish the given
Identify the 𝒓 by using the equation of Absolute Value of Complex Numbers
Plot it on a coordinate plane
To find the angle (known as the argument) Apply the Unit Circle (DO NOT PLUG INTO CALCULATOR WHEN UNIT CIRCLE ANSWERS ARE INVOLVED) by applying the Inverse Tangent function
Apply the appropriate quadrant
Convert to Polar Form equation: 𝒓 𝐜𝐨𝐬 𝜽 +𝒊 𝐬𝐢𝐧 𝜽 or 𝒓 𝐜𝐢𝐬 𝜽
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§6.5: Trig Form of Complex Number

17. §6.5: Trig Form of Complex Number
Example 5
Express 𝟑 +𝒊 into polar/trigonometric form •
𝟑 +𝒊
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§6.5: Trig Form of Complex Number

18. §6.5: Trig Form of Complex Number
Example 5
Express 𝟑 +𝒊 into trigonometric form •
𝟑 +𝒊
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§6.5: Trig Form of Complex Number

19. §6.5: Trig Form of Complex Number
Example 6
Express −𝟒−𝟒𝒊 into trigonometric form
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§6.5: Trig Form of Complex Number

20. §6.5: Trig Form of Complex Number
Your Turn
Express 𝟏−𝒊 into trigonometric form
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§6.5: Trig Form of Complex Number

21. §6.5: Trig Form of Complex Number
Example 7
Express 𝟐−𝟓𝒊 into trigonometric form •
𝟐−𝟓𝒊
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§6.5: Trig Form of Complex Number

22. §6.5: Trig Form of Complex Number
Example 7
Express 𝟐−𝟓𝒊 into trigonometric form •
𝟐−𝟓𝒊
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§6.5: Trig Form of Complex Number

23. §6.5: Trig Form of Complex Number
Example 8
Express −𝟏−𝟒𝒊 into trigonometric form
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§6.5: Trig Form of Complex Number

24. §6.5: Trig Form of Complex Number
Your Turn
Express −𝟐+𝟓𝒊 into trigonometric form
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§6.5: Trig Form of Complex Number

25. §6.5: Trig Form of Complex Number
Assignment
Page odd
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§6.5: Trig Form of Complex Number