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Chapter 13: Problem 44

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\left[\sqrt{3}\left(\cos 70_{1}+i \sin 70_{i}\right)\right]^{3}$$

Short Answer

Expert verified
The rectangular form is \( -\frac{9}{2} - i \frac{3\sqrt{3}}{2} \).

Step by step solution

01

Understand the Problem

The given expression is a complex number in polar form that needs to be raised to the power of 3. It is expressed as \( \left[a(\cos \theta + i \sin \theta)\right]^n \) and we need to find this in rectangular form.
02

Identify Key Components

In the expression \( \left[\sqrt{3}(\cos 70^\circ + i \sin 70^\circ)\right]^3 \), \( a = \sqrt{3} \), \( \theta = 70^\circ \), and \( n = 3 \). We understand to use De Moivre's theorem to expand it and convert into rectangular form.
03

Apply De Moivre's Theorem

De Moivre's Theorem states: \([r (\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta))\). Applying it here, we get: \((\sqrt{3})^3 (\cos(3 \times 70^\circ) + i \sin(3 \times 70^\circ)) = 3\sqrt{3}(\cos 210^\circ + i \sin 210^\circ)\).
04

Evaluate Trigonometric Functions

Now calculate \( \cos 210^\circ \) and \( \sin 210^\circ \). We know \( \cos 210^\circ = -\frac{\sqrt{3}}{2} \) and \( \sin 210^\circ = -\frac{1}{2} \). Replace these values in the expression: \( 3\sqrt{3}(-\frac{\sqrt{3}}{2} + i(-\frac{1}{2})) \).
05

Simplify the Expression

Simplify: \( 3\sqrt{3}( -\frac{\sqrt{3}}{2} + i (-\frac{1}{2})) = 3\sqrt{3} \cdot -\frac{\sqrt{3}}{2} + i \cdot 3\sqrt{3} \cdot -\frac{1}{2} \). This further simplifies to \( -\frac{9}{2} - i \cdot \frac{3\sqrt{3}}{2} \).
06

Express in Rectangular Form

Thus, the expression in rectangular form is \( -\frac{9}{2} - i \frac{3\sqrt{3}}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Form
Complex numbers can be represented in two main forms: rectangular and polar. The polar form is especially useful when dealing with powers and roots of complex numbers. A complex number in polar form is expressed as:
  • \[ z = r(\cos \theta + i\sin \theta) \]
  • Where \( r \) is the modulus of the complex number, giving its magnitude.
  • \( \theta \) is the argument, indicating the angle from the positive x-axis.
With polar form, the calculations for multiplying, dividing, and raising complex numbers to powers become more straightforward. The given problem demonstrates how polar form simplifies exponentiation using De Moivre's theorem.
Rectangular Form
The rectangular form of a complex number is perhaps the most familiar to students. It expresses a complex number as a combination of a real part and an imaginary part:
  • \[ z = a + bi \]
  • Where \( a \) is the real part, and \( b \) is the imaginary part, multiplied by \( i \), the imaginary unit.
In the exercise, we convert the complex number given in polar form back to rectangular form after applying De Moivre's theorem. This involves calculating exact values of trigonometric functions like cosine and sine, leading to a solution like \(-\frac{9}{2} - i \frac{3\sqrt{3}}{2}\). Rectangular form is useful because it directly shows what components make up a complex number.
De Moivre's Theorem
De Moivre's theorem is a powerful tool for dealing with powers of complex numbers, especially when they're in polar form. The theorem states:
  • \[ (r (\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]
  • It allows for straightforward computation of powers and roots of complex numbers.
In the exercise, we used De Moivre's theorem to raise the complex number to the power of three, simplifying the expression to a new angle and modulus in polar form, which we then further converted to rectangular form. Its application illustrates a major advantage of the polar form by simplifying otherwise complex calculations.
Trigonometric Functions
Trigonometric functions like sine and cosine are integral to converting between polar and rectangular forms. They help describe the position of complex numbers on the complex plane:
  • Where cosine corresponds to the x-coordinate (real part).
  • And sine relates to the y-coordinate (imaginary part).
In the exercise, we calculated \( \cos 210^\circ = -\frac{\sqrt{3}}{2} \) and \( \sin 210^\circ = -\frac{1}{2} \). By solving these exact values, we ensured a correct transformation from polar back to rectangular form. Trigonometric functions are fundamental in bridging these two representations, keeping calculations precise and neat.

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Most popular questions from this chapter

Find the indicated roots. Express the results in rectangular form. (a) Compute the four fourth roots of 1 (b) Verify that the sum of these four fourth roots is \(0 .\)

An important model that is used in population biology and ecology is the Ricker model. The Canadian biologist William E. Ricker introduced this model in his paper Stock and Recruitment (Journal of the Fisheries Research Board of Canada, \(11(1954) 559-623\) ). For information on Ricker himself, see the web page The general form of the Ricker model that we will use here is defined by a recursive sequence of the form \(P_{0}=\) initial population at time \(t=0\) \(P_{t}=r P_{t-1} e^{-k P_{t-1}} \quad\) for \(t \geq 1,\) and where \(r\) and \(k\) are positive constants (a) Suppose that the initial size of a population is \(P_{0}=300\) and that the size of the population at the end of year \(t\) is given by$$P_{t}=5 P_{t-1} e^{-P_{t-1} / 1000} \quad(t \geq 1)$$ Use a graphing utility to compute the population sizes through the end of year \(t=5 .\) (As in Example 5, round the final answers to the nearest integers.) Then use the graphing utility to draw the population scatter plot for \(t=0,1, \ldots, 5 .\) Describe in complete sentences how the size of the population changes over this period. Does the population seem to be approaching an equilibrium level? (b) Using a graphing utility, compute the sizes of the population in part (a) through the end of the year \(t=20\) and draw the corresponding scatter plot. Note that the population seems to be approaching an equilibrium level of \(1609(\text { or } 1610)\) (c) Determine the equilibrium population algebraically by solving the following equation for \(P_{t-1} .\) For the final answer, use a calculator and round to the nearest integer. $$P_{t-1}=5 P_{t-1} e^{-P_{t-1} / 1000}$$

Find the indicated roots. Express the results in rectangular form. Evaluate \(\left(-\frac{1}{2}+\frac{1}{2} \sqrt{3} i\right)^{5}+\left(-\frac{1}{2}-\frac{1}{2} \sqrt{3} i\right)^{5} .\) Hint: Use DeMoivre's theorem.

Use a calculator to complete Compute the cube roots of \(1+2 i\). Express your answers in rectangular form, with the real and imaginary parts rounded to two decimal places.

Express each of the sums without using sigma notation. Simplify your answers where possible. $$\sum_{j=1}^{5}\left(x^{j+1}-x^{j}\right)$$
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