How do I find the power of (3-i)^12 using trigonometric form?

In summary, a complex number is a number that contains both a real part and an imaginary part, and can be written in the form a+bi. The power of a complex number is the result of raising that number to a certain exponent, using the same rules as for real numbers. To calculate the power of a complex number, one can use the polar or rectangular form and De Moivre's theorem. Complex numbers have many applications in science, including in physics, engineering, and mathematics. They can also have a negative power, resulting in a reciprocal of the original number, but with potentially different magnitude and angle.
  • #1
hellbike
61
0
how to find number [tex](3-i)^{12}[/tex] ?

i know theorem for powers of complex numbers, but i have to know argument of trygonometrical form.

and its [tex] cos x = 3/4[/tex] and [tex]sin x = -1/4[/tex]

and i don't know what to do with that.
 
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  • #2
First write [itex]3 - i [/itex] in the form [itex]r (\cos\theta + i \sin \theta)[/itex] . Then [itex](3 - i)^{12} [/itex] will be [itex]r^{12} (\cos(12 \theta) + i \sin (12 \theta))[/itex]
 

FAQ: How do I find the power of (3-i)^12 using trigonometric form?

What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It can be written in the form a+bi, where a is the real part and bi is the imaginary part (with i representing the square root of -1).

What is the power of a complex number?

The power of a complex number is the result of raising that number to a certain exponent. This is done by using the same rules as for real numbers, with the added consideration of the imaginary unit i.

How do you calculate the power of a complex number?

To calculate the power of a complex number, you can use the polar form or the rectangular form of the number. In both cases, you can use the De Moivre's theorem to simplify the calculation.

What are the applications of complex numbers in science?

Complex numbers have many applications in science, including in physics, engineering, and mathematics. They are used to describe and analyze phenomena that involve oscillations, such as waves and electricity, and are also used in the study of quantum mechanics and signal processing.

Can a complex number have a negative power?

Yes, a complex number can have a negative power. This is because complex numbers have both a real and imaginary part, and raising a complex number to a negative power will result in a reciprocal of the original number. However, the result may be a complex number with a different magnitude and angle.

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