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**Math 112 Elementary Functions**

Chapter 7 – Applications of Trigonometry Section 3 Complex Numbers: Trigonometric Form

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**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. 1 y x x is a positive real number y is a negative real number

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**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)

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**Graphing Complex Numbers**

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

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**Graphing Complex Numbers**

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

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**Graphing Complex Numbers**

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

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**|x| = distance from the origin**

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

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**|a + bi| = distance from the origin**

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

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**Trigonometric Form of a Complex Number**

a + bi r a b Therefore, a + bi = r (cos + i sin) Note: As a standard, is to be the smallest positive number possible.

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**Trigonometric Form of a Complex Number**

Example: – 3i Steps for finding the trig form of a + bi. r = |a + bi| is determined by … cos = a / r sin = b / r

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**Trigonometric Form of a Complex 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.

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**Trigonometric Form of Real and Imaginary Numbers (examples)**

Real/Imaginary Number Complex Form Trig w/ Degrees Radians 0 + 0i 0 cis 0° 0 cis 0 2 2 + 0i 2 cis 0° 2 cis 0 -5 -5 + 0i 5 cis 180° 2 cis 3i 0 + 3i 3 cis 90° 3 cis (/2) -4i 0 – 4i 4 cis 270° 4 cis (3/2)

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**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 = 23 + 2i i

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**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!

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**Multiplication of Complex Numbers (Standard Form)**

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**Multiplication of Complex Numbers (Trigonometric Form)**

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**Division of Complex Numbers (Standard Form)**

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**Division of Complex Numbers (Trigonometric Form)**

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**Powers of Complex Numbers (Trigonometric Form)**

[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

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**Powers of Complex Numbers (Trigonometric Form)**

DeMoivre’s Theorem (r cis )n = rn cis (n)

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**Roots of Complex Numbers**

An nth root of a number (a+bi) is any solution to the equation … xn = a+bi

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**Roots of Complex Numbers**

Examples The two 2nd roots of 9 are … 3 and -3, because: = and (-3)2 = 9 The two 2nd roots of -25 are … 5i and -5i, because: (5i)2 = and (-5i)2 = -25 The two 2nd roots of 16i are … 22 + 22i and 2 - 22i because (22 + 22i)2 = 16i and (-22 - 22i)2 = 16i

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**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

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**Roots of Complex Numbers**

In general, there are always … n “nth roots” of any complex number

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**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.

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**Roots of Complex Numbers**

The n nth roots of the complex number r(cos + i sin ) are …

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**Roots of Complex Numbers**

The n nth roots of the complex number r cis are … or

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**Summary of (r cis ) w/ r = 1**

} Does this remind you of something?

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**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/r sin = b/r

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**Results of Euler’s Formula**

This gives a relationship between the 4 most common constants in mathematics!

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**Results of Euler’s Formula**

ii is a real number!

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**Results of Euler’s Formula**

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