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Copyright © 2011 Pearson Education, Inc. Slide Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representations Call the horizontal axis the real axis and the vertical axis the imaginary axis. Now complex numbers can be graphed in this complex plane. The sum of two complex numbers can be represented graphically by the vector that is the resultant of the sum of vectors corresponding to the two numbers.

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Copyright © 2011 Pearson Education, Inc. Slide Expressing the Sum of Complex Numbers Graphically ExampleFind the sum of 6 – 2i and –4 – 3i. Graph both complex numbers and their resultant. Solution (6 – 2i) + (–4 – 3i) = 2 – 5i

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Copyright © 2011 Pearson Education, Inc. Slide Trigonometric (Polar) Form Relationship Among x, y, r, and The graph shows the complex number x + yi that corresponds to the vector OP.

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Copyright © 2011 Pearson Education, Inc. Slide Trigonometric (Polar) Form Substituting x = r cos and y = r sin into x + yi gives Trigonometric or Polar Form of a Complex Number The expression r(cos + i sin ) is called the trigonometric form (or polar form) of the complex number x + yi.

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Copyright © 2011 Pearson Education, Inc. Slide Trigonometric (Polar) Form Notation: cos + i sin is sometimes written cis. Using this notation, r(cos + i sin ) is written r cis. The number r is called the modulus or absolute value of the complex number x + yi. Angle is called the argument of the complex number x + yi.

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Copyright © 2011 Pearson Education, Inc. Slide Converting from Trigonometric Form to Rectangular Form ExampleExpress 2(cos 300º + i sin 300º) in rectangular form. Analytic Solution Graphing Calculator Solution

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Copyright © 2011 Pearson Education, Inc. Slide Converting from Rectangular to Trigonometric Form Converting from Rectangular to Trigonometric Form 1.Sketch a graph of the number x + yi in the complex plane. 2.Find r by using the equation 3.Find by using the equation tan = y/x, x 0, choosing the quadrant indicated in Step 1.

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Copyright © 2011 Pearson Education, Inc. Slide Converting from Rectangular to Trigonometric Form ExampleWrite each complex number in trigonometric form. Solution (a)Start by sketching the graph of in the complex plane. Then find r.

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Copyright © 2011 Pearson Education, Inc. Slide Converting from Rectangular to Trigonometric Form Now find. Therefore, in polar form, is in quadrant II and tan = the reference angle in quadrant II is

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Copyright © 2011 Pearson Education, Inc. Slide Converting from Rectangular to Trigonometric Form (b) From the graph, = 270º. In trigonometric form, different way to determine.

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Copyright © 2011 Pearson Education, Inc. Slide Products of Complex Numbers in Trigonometric Form Multiplying complex numbers in rectangular form. Multiplying complex numbers in trigonometric form.

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Copyright © 2011 Pearson Education, Inc. Slide Products of Complex Numbers in Trigonometric Form Product Theorem If are any two complex numbers, then In compact form, this is written

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Copyright © 2011 Pearson Education, Inc. Slide Using the Product Theorem ExampleFind the product of 3(cos 45º + i sin 45º) and 2(cos 135º + i sin 135º). Solution

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Copyright © 2011 Pearson Education, Inc. Slide Quotients of Complex Numbers in Trigonometric Form The rectangular form of the quotient of two complex numbers. The polar form of the quotient of two complex numbers.

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Copyright © 2011 Pearson Education, Inc. Slide Quotients of Complex Numbers in Trigonometric Form Quotient Theorem If r 1 (cos 1 + i sin 1 ) and r 2 (cos 2 + i sin 2 ) are complex numbers, where r 2 (cos 2 + i sin 2 ) 0, then In compact form, this is written

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Copyright © 2011 Pearson Education, Inc. Slide Using the Quotient Theorem ExampleFind the quotient Write the result in rectangular form. Solution

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