M1410605 - (Section 6.5: Trig and Euler Forms of a Complex...

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(Section 6.5: Trig and Euler Forms of a Complex Number) 6.33 SECTION 6.5: TRIG (AND EULER / EXPONENTIAL) FORMS OF A COMPLEX NUMBER See the Handout on my website . PART A: DIFFERENT FORMS OF A COMPLEX NUMBER Let a , b , and r be real numbers, and let θ be measured in either degrees or radians (in which case it could be treated as a real number.) We often let z denote a complex number. Standard or Rectangular Form: z = a + bi We saw this form in Section 2.4 . The complex number a + bi may be graphed in the complex plane as either the point a , b ( ) or as the position vector a , b , in which case our analyses from Section 6.3 become helpful. Trig Form: z = r cos + i sin ( ) r takes on the role of v from our discussion of vectors. It is the distance of the point representing the complex number from 0. has a role similar to the one it had in our discussion of vectors, namely as a direction angle, but this time in the complex plane. This is derived from the Standard Form through the relations: a = r cos , and b = r sin Recall from Section 6.3 , with r replacing v : r = a 2 + b 2 Choose such that: tan = b a if a 0 ( ) , and is in the correct Quadrant
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(Section 6.5: Trig and Euler Forms of a Complex Number) 6.34 Euler (Exponential) Form: z = re i θ This is useful to derive various formulas in the Handout . In this class, you will not be required to use this form.
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410605 - (Section 6.5: Trig and Euler Forms of a Complex...

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