QuestionCalculate the power and express it in rectangular form: $[5(\cos 30^{\circ}+i \sin 30^{\circ})]^{4}$ .

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in polar form $5(\cos30^{\circ}+i \sin30^{\circ})$ . . We are asked to find the fourth power of this complex number.
3. The result should be written in rectangular form.

STEP 2

First, we need to use De Moivre's theorem to find the power of a complex number in polar form. De Moivre's theorem states that if $z=r(\cos \theta +i \sin \theta)$ , then $z^n=r^n(\cos n\theta +i \sin n\theta)$ .
So, we have $\left[5\left(\cos30^{\circ}+i \sin30^{\circ}\right)\right]^{4} =5^4 \left[\cos \left(4 \times30^{\circ}\right)+i \sin \left(4 \times30^{\circ}\right)\right]$

STEP 3

Now, calculate the values.
5^ \left[\cos \left( \times30^{\circ}\right)+i \sin \left( \times30^{\circ}\right)\right] =625 \left[\cos120^{\circ}+i \sin120^{\circ}\right]

STEP 4

We know that $\cos120^{\circ} = -\frac{1}{2}$ and $\sin120^{\circ} = \frac{\sqrt{3}}{2}$ .
So, we have $625 \left[\cos120^{\circ}+i \sin120^{\circ}\right] =625 \left[-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]$

STEP 5

Now, multiply the real and imaginary parts by625 to convert the complex number to rectangular form.
$625 \left[-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right] = -\frac{625}{2}+i \frac{625\sqrt{3}}{2}$

STEP 6

implify the expression.
$-\frac{625}{2}+i \frac{625\sqrt{3}}{2} = -312.5 + i312.5\sqrt{3}$ So, the fourth power of the given complex number in rectangular form is $-312.5 + i312.5\sqrt{3}$ .

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