Trigonometric Form Section 6.5 Precalculus PreAP/Dual, Revised ©2016 Viet.dang@humbleisd.net 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Review Solve 𝒚= 𝒙 𝟐 −𝟏 and graph on a coordinate plane 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Review Solve 𝒚= 𝒙 𝟐 +𝟏 and graph on a coordinate plane 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number The Complex Plane Graphing the Complex Plane Imaginary Axis Real Axis O 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 1 Graph each complex number: 𝟐−𝟑𝒊 • 2 – 3i 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 2 Graph each complex number: 𝟐−𝒊 𝟐+𝒊 • 5 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Your Turn Graph each complex number: 𝟐𝒊 𝟐−𝒊 • 2 + 4i 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

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 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number The Complex Plane Graphing the Complex Plane Imaginary Axis Real Axis O 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 3 Solve |𝟑+𝟒𝒊| 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 4 Solve −𝟐+𝟒𝒊 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Your Turn Solve |𝟏𝟎−𝟒𝒊| 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Trigonometric Form Identify the relationship of 𝒙, 𝒚, 𝒓, and 𝜽 in terms of trig functions 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Trigonometric Form Identify the relationship of 𝒙, 𝒚, 𝒓, and 𝜽 in terms of trig functions 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§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 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

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 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 5 Express 𝟑 +𝒊 into polar/trigonometric form • 𝟑 +𝒊 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 5 Express 𝟑 +𝒊 into trigonometric form • 𝟑 +𝒊 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 6 Express −𝟒−𝟒𝒊 into trigonometric form 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Your Turn Express 𝟏−𝒊 into trigonometric form 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 7 Express 𝟐−𝟓𝒊 into trigonometric form • 𝟐−𝟓𝒊 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 7 Express 𝟐−𝟓𝒊 into trigonometric form • 𝟐−𝟓𝒊 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Example 8 Express −𝟏−𝟒𝒊 into trigonometric form 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Your Turn Express −𝟐+𝟓𝒊 into trigonometric form 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number

§6.5: Trig Form of Complex Number Assignment Page 448 5-29 odd 2/28/2019 10:12 AM §6.5: Trig Form of Complex Number