Trigonometry Lecture Notes_part3

# Solution the complex number z is in polar form with r

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Solution: The complex number z is in polar form, with r = 4 and 30 θ = ° . All we have to do is to evaluate the trigonometric functions in Z to get the rectangular form. Thus ( ) 3 1 4 cos30 sin30 4 2 3 2 2 2 z i i i = °+ ° = + = +

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Example 121
Example 122 Find the quotient of the complex numbers and leave your answer in polar form: 1 4 4 50 cos sin 3 3 z i π π = + and 2 5 cos sin 3 3 z i π π = +

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Example 123 Example 124 Find ( ) 8 1 i + and write your answer in rectangular form.
Example 125 Find all the complex fourth roots of 81(cos60º+isin60º) 4 360 360 cos sin 60 360*0 60 360*0 81 cos sin 4 4 3(cos15 sin15 ) n k k k z r i n n i i θ θ + + = + + + = + = + ringoperator ringoperator 4 60 360*1 60 360*1 81 cos sin 4 4 3(cos105 sin105 ) 3(cos195 sin195 ) 3(cos285 sin 285 ) i i i i + + = + = + = + = + ringoperator ringoperator ringoperator ringoperator ringoperator ringoperator

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Example 126 Find all of the cube roots of 8 and express your answers in rectangular form. Solution: Since DeMoivre’s Theorem applies for the roots of complex numbers in polar form, we need to first write 8 into polar form. ( ) ( ) 8 cos sin 8 cos0 sin 0 r i i θ θ = + = °+ ° 3 0 0 2 *0 0 2 *0 8 cos sin 3 3 z i π π + + = + 3 1 0 2 *1 0 2 *1 8 cos sin 3 3 z i π π + + = + 3 2 0 2 *2 0 2 *2 8 cos sin 3 3 z i π π + + = + Section 8.3 Vectors
Example 127 Vector U has initial point ( -3, -3) and terminal point (0, 3). Vector V has initial point ( 0, 0) and terminal point ( 3, 6). Show that vectors V and U are equal.

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