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Math 112 Elementary Functions

Math 112 Elementary Functions. Chapter 7 – Applications of Trigonometry. Section 3 Complex Numbers: Trigonometric Form. y. 0. 1. x. y is a negative real number. x is a positive real number. Graphing Complex Numbers. How do you graph a real number? Use a number line.

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Math 112 Elementary Functions

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  1. Math 112Elementary Functions Chapter 7 – Applications of Trigonometry Section 3 Complex Numbers: Trigonometric Form

  2. y 0 1 x y is a negative real number x is a positive real number Graphing Complex Numbers • How do you graph a real number? • Use a number line. • The point corresponding to a real number represents the directed distance from 0.

  3. Graphing Complex Numbers • General form of a complex number … a + bi • a  R and b  R • i = -1 Therefore, a complex number is essentially an ordered pair! (a, b)

  4. -4 2 Graphing Complex Numbers Imaginary Axis Real Axis All real numbers, a = a+0i, lie on the real axis at (a, 0).

  5. 2i -4i Graphing Complex Numbers Imaginary Axis All imaginary numbers, bi = 0+bi, lie on the imaginary axis at (0, b). Real Axis

  6. 2 + 3i -4 + i 3 – 2i -3 - 4i Graphing Complex Numbers Imaginary Axis Real Axis All other numbers, a+bi, are located at the point (a,b).

  7. Absolute Value • Real Numbers: |x| = distance from the origin

  8. a + bi b a Absolute Value • Complex Numbers: |a + bi| = distance from the origin Note that if b = 0, then this reduces to an equivalent definition for the absolute value of a real number.

  9. a + bi r b a Trigonometric Form of aComplex Number  Therefore, a + bi = r (cos + i sin) Note: As a standard,  is to be the smallest positive number possible.

  10. Trigonometric Form of aComplex Number Example: 2 – 3i • Steps for finding the trig form of a + bi. • r = |a + bi| •  is determined by … cos  = a / r sin  = b / r

  11. Trigonometric Form of aComplex Number – Determining  • a + bi = r cis  r = |a+bi| cos  = a/r sin  = b/r • Using cos  = a/r • Q1:  = cos-1(a/r) • Q2:  = cos-1(a/r) • Q3:  = 360° - cos-1(a/r) • Q4:  = 360° - cos-1(a/r) • Using sin  = b/r • Q1:  = sin-1(b/r) • Q2:  = 180° - sin-1(b/r) • Q3:  = 180° - sin-1(b/r) • Q4:  = 360° + sin-1(b/r) For Radians, replace 180° with  and 360° with 2.

  12. Trigonometric Form of Real and Imaginary Numbers (examples)

  13. Converting the Trigonometric Form to Standard Form • r cis  = r (cos  + i sin ) = (r cos ) + (r sin ) i • Example: 4 cis 30º = (4 cos 30º) + (4 sin 30º)i = 4(3)/2 + 4(1/2)i = 23 + 2i  3.46 + 2i

  14. Arithmetic with Complex Numbers • Addition & Subtraction • Standard form is very easy ………Trig. form is ugly! • Multiplication & Division • Standard form is ugly…………….Trig. form is easy! • Exponentiation & Roots • Standard form is very ugly….Trig. form is very easy!

  15. Multiplication of Complex Numbers(Standard Form)

  16. Multiplication of Complex Numbers(Trigonometric Form)

  17. Division of Complex Numbers(Standard Form)

  18. Division of Complex Numbers(Trigonometric Form)

  19. [r cis ]2 = (r cis ) • (r cis ) = r2 cis( + ) = r2 cis 2 [r cis ]3 = (r cis )2• (r cis ) = r2 cis(2) •(r cis ) = r3 cis 3 Powers of Complex Numbers(Trigonometric Form)

  20. Powers of Complex Numbers(Trigonometric Form) • DeMoivre’s Theorem (r cis )n = rn cis (n)

  21. Roots of Complex Numbers • An nth root of a number (a+bi) is any solution to the equation … xn = a+bi

  22. Roots of Complex Numbers • Examples • The two 2nd roots of 9 are … • 3 and -3, because: 32 = 9 and (-3)2 = 9 • The two 2nd roots of -25 are … • 5i and -5i, because: (5i)2 = -25 and (-5i)2 = -25 • The two 2nd roots of 16i are … • 22 + 22i and -22 - 22i because (22 + 22i)2 = 16i and (-22 - 22i)2 = 16i

  23. Roots of Complex Numbers • Example: Find all of the 4th roots of 16. • x4 = 16 • x4 – 16 = 0 • (x2 + 4)(x2 – 4) = 0 • (x + 2i)(x – 2i)(x + 2)(x – 2) = 0 • x = ±2i or ±2

  24. Roots of Complex Numbers • In general, there are always … n “nth roots” of any complex number

  25. Roots of Complex Numbers • One more example … Using DeMoivre’s Theorem Let k = 0, 1, & 2 NOTE: If you let k = 3, you get 2cis385 which is equivalent to 2cis25.

  26. Roots of Complex Numbers The n nth roots of the complex number r(cos  + i sin ) are …

  27. Roots of Complex Numbers The n nth roots of the complex number r cis  are … or

  28. } Does this remind you of something? Summary of (r cis ) w/ r = 1

  29. Euler’s Formula Note:  must be expressed in radians. Therefore, the complex number … r = |a + bi| cos  = a/r sin  = b/r

  30. Results of Euler’s Formula This gives a relationship between the 4 most common constants in mathematics!

  31. Results of Euler’s Formula ii is a real number!

  32. Results of Euler’s Formula

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