Transcription
Good morning. Today we are going to see how to calculate the power of a complex number. In our case, our complex number will be represented by two plus two i, and this will be equal to z. We will have to calculate when this number is raised to the power of eight. For this, we will use this De Moivre's theorem, which tells us that if we have our complex number z equal to r times the cosine of theta plus i times the sine of theta, then z to the power of n, where n represents our power, will be equal to r, the modulus, to the power of n, multiplied by the cosine of n times theta plus i times the sine of n times theta.
As we see, De Moivre's theorem requires us to have the number in polar form, also called cis. In this case, we have the number in binomial form, which is represented in the form a plus i times b, where a is the real component and b is the imaginary component. In our case, a would be equal to 2, and b, which we always recognize because it is accompanied by i, would also be equal to 2.
We will represent this in our complex plane. We have a component equal to 2 on the real axis and a component equal to 2 on the imaginary axis. Our complex vector would have coordinates 2, 2, and the vector would be drawn in this way. The value of r refers to the modulus of the vector, or the hypotenuse of this right-angled triangle that has been formed, taking into account the components on the real axis and the imaginary axis, and the angle formed by the real axis and the complex vector.
We will find r using the Pythagorean theorem, considering that r is the hypotenuse of this right-angled triangle. R, also called the modulus, is equal to the square root of each of the components of this vector squared. The modulus of our vector will be equal to the square root of 2 squared, which is 4, plus 2 squared, which is 4. This would be equal to the square root of 8, and we leave it this way since the square root of 8 is not a perfect square.
To find our angle, knowing the opposite side and the adjacent side, we can use any of the trigonometric functions like cosine or sine. However, in this type of exercise, it is very common to use the tangent of the angle, which is equal to the opposite side over the adjacent side, or what is exactly the same, the component in i, that is, the imaginary part over the real part. Solving for this, the angle will be equal to the arctangent of b over a. Our angle will be equal to 45 degrees.
Knowing the angle and knowing the modulus, we can now express our complex number, which we initially had in binomial form, now in polar form. It would be that z is equal to the modulus, which is the square root of 8, times the cosine of the angle 45 degrees plus the sine of 45 degrees.
Now we will calculate z to the power of 8 according to De Moivre's formula. We have to take the modulus and raise it to the power we are looking for. In this case, it will be the square root of 8 to the power of 8, multiplied by the cosine of 8 times 45 degrees, plus i times the sine of 8 times 45 degrees. Z to the power of 8 will be equal to the square root of 8 to the power of 8. This is equal to 4096. Cosine of 8 times 45 degrees is equal to the cosine of 360 degrees, plus i times the sine of 360 degrees. The cosine of 360 degrees is equal to 1, and the sine of 360 degrees is equal to 0. So, i times 0 would cancel out this expression, and we would be left with z to the power of 8 being equal to 4096 times 1, meaning that when calculating the eighth power of our complex number, it will be equal to 4096.
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