Presentation on theme: "Math 112 Elementary Functions"— Presentation transcript:
1 Math 112 Elementary Functions Chapter 7 – Applications of TrigonometrySection 3Complex Numbers: Trigonometric Form
2 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 0.1yxx is a positive real numbery is a negative real number
3 Graphing Complex Numbers General form of a complex number …a + bia R and b Ri = -1Therefore, a complex number is essentially an ordered pair!(a, b)
4 Graphing Complex Numbers Imaginary Axis-42Real AxisAll real numbers, a = a+0i, lie on the real axis at (a, 0).
5 Graphing Complex Numbers Imaginary AxisAll imaginary numbers, bi = 0+bi, lie on the imaginary axis at (0, b).2iReal Axis-4i
6 Graphing Complex Numbers Imaginary Axis2 + 3i-4 + iReal Axis3 – 2i-3 - 4iAll other numbers, a+bi, are located at the point (a,b).
7 |x| = distance from the origin Absolute ValueReal Numbers:|x| = distance from the origin
8 |a + bi| = distance from the origin Absolute ValueComplex Numbers:|a + bi| = distance from the origina + biabNote that if b = 0, then this reduces to an equivalent definition for the absolute value of a real number.
9 Trigonometric Form of a Complex Number a + birabTherefore,a + bi = r (cos + i sin)Note: As a standard, is to be the smallest positive number possible.
10 Trigonometric Form of a Complex Number Example: – 3iSteps for finding the trig form of a + bi.r = |a + bi| is determined by …cos = a / rsin = b / r
11 Trigonometric Form of a Complex Number – Determining a + bi = r cis r = |a+bi| cos = a/r sin = b/rUsing cos = a/rQ1: = cos-1(a/r)Q2: = cos-1(a/r)Q3: = 360° - cos-1(a/r)Q4: = 360° - cos-1(a/r)Using sin = b/rQ1: = 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.
13 Converting the Trigonometric Form to Standard Form r cis = r (cos + i sin )= (r cos ) + (r sin ) iExample: 4 cis 30º= (4 cos 30º) + (4 sin 30º)i= 4(3)/2 + 4(1/2)i= 23 + 2i i
14 Arithmetic with Complex Numbers Addition & SubtractionStandard form is very easy ………Trig. form is ugly!Multiplication & DivisionStandard form is ugly…………….Trig. form is easy!Exponentiation & RootsStandard form is very ugly….Trig. form is very easy!
15 Multiplication of Complex Numbers (Standard Form)
16 Multiplication of Complex Numbers (Trigonometric Form)
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 ExamplesThe two 2nd roots of 9 are …3 and -3, because: = and (-3)2 = 9The two 2nd roots of -25 are …5i and -5i, because: (5i)2 = and (-5i)2 = -25The two 2nd roots of 16i are …22 + 22i and 2 - 22ibecause (22 + 22i)2 = 16i and (-22 - 22i)2 = 16i
23 Roots of Complex Numbers Example:Find all of the 4th roots of 16.x4 = 16x4 – 16 = 0(x2 + 4)(x2 – 4) = 0(x + 2i)(x – 2i)(x + 2)(x – 2) = 0x = ±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 TheoremLet k = 0, 1, & 2NOTE: 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 Summary of (r cis ) w/ r = 1 }Does this remind youof something?
29 Euler’s Formula Therefore, the complex number … r = |a + bi| Note: must be expressed in radians.Therefore, the complex number …r = |a + bi|cos = a/rsin = 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!