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**10.4 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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**10.4 Expressing the Sum of Complex Numbers Graphically**

Example Find 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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**10.4 Trigonometric (Polar) Form**

The graph shows the complex number x + yi that corresponds to the vector OP. Relationship Among x, y, r, and

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**10.4 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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**10.4 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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**10.4 Converting from Trigonometric Form to Rectangular Form**

Example Express 2(cos 300º + i sin 300º) in rectangular form. Analytic Solution Graphing Calculator Solution

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**10.4 Converting from Rectangular to Trigonometric Form**

Sketch a graph of the number x + yi in the complex plane. Find r by using the equation Find by using the equation tan = y/x, x 0, choosing the quadrant indicated in Step 1.

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**10.4 Converting from Rectangular to Trigonometric Form**

Example Write each complex number in trigonometric form. Solution Start by sketching the graph of in the complex plane. Then find r.

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**10.4 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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**10.4 Converting from Rectangular to Trigonometric Form**

(b) From the graph, = 270º. In trigonometric form, different way to determine .

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**10.4 Products of Complex Numbers in Trigonometric Form**

Multiplying complex numbers in rectangular form. Multiplying complex numbers in trigonometric form.

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**10.4 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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**10.4 Using the Product Theorem**

Example Find the product of 3(cos 45º + i sin 45º) and 2(cos 135º + i sin 135º). Solution

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**10.4 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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**10.4 Quotients of Complex Numbers in Trigonometric Form**

Quotient Theorem If r1(cos 1 + i sin 1) and r2(cos 2 + i sin 2) are complex numbers, where r2(cos 2 + i sin 2) 0, then In compact form, this is written

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**10.4 Using the Quotient Theorem**

Example Find the quotient Write the result in rectangular form. Solution

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Applications of Trigonometric Functions

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