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Chapter 10: Applications of Trigonometry and Vectors

Chapter 10: Applications of Trigonometry and Vectors. 10.1 The Law of Sines 10.2 The Law of Cosines and Area Formulas 10.3 Vectors and Their Applications 10.4 Trigonometric (Polar) Form of Complex Numbers 10.5 Powers and Roots of Complex Numbers 10.6 Polar Equations and Graphs

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Chapter 10: Applications of Trigonometry and Vectors

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  1. Chapter 10: Applications of Trigonometry and Vectors 10.1 The Law of Sines 10.2 The Law of Cosines and Area Formulas 10.3 Vectors and Their Applications 10.4 Trigonometric (Polar) Form of Complex Numbers 10.5 Powers and Roots of Complex Numbers 10.6 Polar Equations and Graphs 10.7 More Parametric Equations

  2. 10.4 Trigonometric (Polar) Form of Complex Numbers 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 Complex Plane and Vector Representations 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.

  3. 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

  4. 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 

  5. 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 polarform) of the complex number x + yi.

  6. 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.

  7. 10.4 Converting from Trigonometric Form to Rectangular Form Example Express2(cos 300º + i sin 300º) in rectangular form. Analytic Solution Graphing Calculator Solution

  8. 10.4 Converting from Rectangular to Trigonometric Form • 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.

  9. 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.

  10. 10.4 Converting from Rectangular to Trigonometric Form  is in quadrant II and tan  = the reference angle in quadrant II is Now find . Therefore, in polar form,

  11. 10.4 Converting from Rectangular to Trigonometric Form (b) From the graph,  = 270º. In trigonometric form, different way to determine .

  12. 10.4 Deciding Whether a Number is in the Julia Set Example The fractal called the Julia set is shown in the figure. To determine if a complex number z = a + bi is in this Julia set, perform the following sequence of calculations. Repeatedly compute the values of z2 – 1, (z2 – 1)2 –1, [(z2 – 1)2 –1]2 – 1, . . . . If the moduli of any of the resulting complex numbers exceeds 2, then z is not in the Julia set. Otherwise z is part of this set and the point (a, b) should be shaded in the graph.

  13. 10.4 Deciding Whether a Number is in the Julia Set Determine if z = 0 + 0i belongs to the Julia set. Solution So, and so on. The calculations repeat as 0, –1, 0, –1, and so on. The moduli are either 0 or 1, therefore, 0 + 0i belongs to the Julia set.

  14. 10.4 Products of Complex Numbers in Trigonometric Form • Multiplyingcomplex numbers in rectangular form. • Multiplying complex numbers in trigonometric form.

  15. 10.4 Products of Complex Numbers in Trigonometric Form Product Theorem If are any two complex numbers, then In compact form, this is written

  16. 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

  17. 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.

  18. 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

  19. 10.4 Using the Quotient Theorem Example Find the quotient Write the result in rectangular form. Solution

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