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Find [5(cos 330 degrees + I sin 330 degrees)]^3

Find [5(cos 330 degrees + I sin 330 degrees)]^3-example-1
User Mustafa J
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1 Answer

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Given a complex number in the form:

z= \rho [\cos \theta + i \sin \theta]
The nth-power of this number,
z^n, can be calculated as follows:

- the modulus of
z^n is equal to the nth-power of the modulus of z, while the angle of
z^n is equal to n multiplied the angle of z, so:

z^n = \rho^n [\cos n\theta + i \sin n\theta ]
In our case, n=3, so
z^3 is equal to

z^3 = \rho^3 [\cos 3 \theta + i \sin 3 \theta ] = (5^3) [\cos (3 \cdot 330^(\circ)) + i \sin (3 \cdot 330^(\circ)) ] (1)
And since

3 \cdot 330^(\circ) = 990^(\circ) = 2\pi +270^(\circ)
and both sine and cosine are periodic in
2 \pi, (1) becomes

z^3 = 125 [\cos 270^(\circ) + i \sin 270^(\circ) ]

User Justin Wrobel
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