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Chapter 10: Problem 33

Write each complex number in rectangular form. $$ 0.2\left(\cos \frac{5 \pi}{9}+i \sin \frac{5 \pi}{9}\right) $$

Short Answer

Expert verified
-0.03472 + 0.19696i

Step by step solution

01

- Identify the given complex number in polar form

The given complex number is presented in polar form as \(0.2\left(\cos \frac{5 \pi}{9} + i \sin \frac{5 \pi}{9}\right)\). In this form, \(r = 0.2\), \(\theta = \frac{5 \pi}{9}\).
02

- Convert the polar form to rectangular form

Use the conversion formulas: \(z = r(\cos \theta + i \sin \theta)\). Substitute \(r = 0.2\) and \(\theta = \frac{5 \pi}{9}\) into the formulas for cosine and sine.
03

- Calculate \(\cos \frac{5 \pi}{9}\)

Using a calculator, find that \(\cos \frac{5 \pi}{9} \approx -0.1736\).
04

- Calculate \(\sin \frac{5 \pi}{9}\)

Using a calculator, find that \(\sin \frac{5 \pi}{9} \approx 0.9848\).
05

- Substitute back into the rectangular coordinates

Substitute the values found in steps 3 and 4 to get: \(0.2\left(\cos \frac{5 \pi}{9} + i \sin \frac{5 \pi}{9}\right) = 0.2(-0.1736 + 0.9848i)\).
06

- Perform the multiplication

Multiply \(0.2\) by each part: $$0.2(-0.1736) + 0.2(0.9848i)= -0.03472 + 0.19696i.$$. Thus, the rectangular form of the given complex number is $$-0.03472 + 0.19696i.$$

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Key Concepts

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

Polar to Rectangular Conversion
To understand complex numbers better, we often switch between their polar and rectangular forms. Each form offers unique insights and simplifications for different scenarios. Let's look at how to convert from polar to rectangular form. In polar form, a complex number is expressed as: \[ z = r(\text{cos } \theta + i \text{ sin } \theta) \]Here,
  • r: Magnitude of the complex number
  • \(\theta\): Angle with the positive real axis
Conversion involves substituting the known values of \( r \) and \( \theta \) into the equations for cosine and sine. After calculating these trigonometric values, multiply them by \( r \) to find the rectangular form, which is expressed as: \[ z = a + bi \]This form provides the real part \( a \) and the imaginary part \( b \) separately.
Cosine Function
In a polar to rectangular conversion, the cosine function plays a crucial role. The cosine of an angle \(\theta\) in a right triangle defines the ratio of the adjacent side to the hypotenuse: \[ \text{cos } \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]However, in our case, it's often easier to calculate cos using a calculator. For the given angle, \( \frac{5\pi}{9} \), using a calculator we find: \[ \text{cos } \frac{5\pi}{9} \thickapprox -0.1736 \]This value is then used to find the real part in the rectangular form. It's important to note that cosines of certain angles can be negative depending on the quadrant in which the angle lies.
Sine Function
The sine function is equally important in these conversions. Similar to cosine, sine helps define the imaginary part of a complex number. For an angle \( \theta \), sine is the ratio of the opposite side to the hypotenuse: \[ \text{sin } \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \]For the angle \( \frac{5\pi}{9} \), using a calculator, we get: \[ \text{sin } \frac{5\pi}{9} \thickapprox 0.9848 \]This value is then multiplied by the magnitude \( r \) to find the imaginary part in rectangular form. In the trigonometric circle, sine values will always be positive or negative based on the angle's positioning.
Mathematical Calculations
Finally, after deriving the exact values for cosine and sine, we perform some basic multiplications: \[ z = r(\text{cos} \theta + i \text{sin} \theta) \]Substitute \( r = 0.2 \), \( \text{cos} \frac{5\pi}{9} = -0.1736 \), and \( \text{sin} \frac{5\pi}{9} = 0.9848 \): \[ z = 0.2(-0.1736 + 0.9848i) \]This simplifies to: \[ z \thickapprox -0.03472 + 0.19696i \]Thus, the rectangular form is \[ -0.03472 + 0.19696i \]This step aligns our results accurately, verifying that both the magnitude and angle contribute correctly to the complex number's real and imaginary parts.

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Most popular questions from this chapter

A helicopter pilot needs to travel to a regional airport 25 miles away. She flies at an actual heading of \(\mathrm{N} 16.26^{\circ} \mathrm{E}\) with an airspeed of \(120 \mathrm{mph},\) and there is a wind blowing directly east at \(20 \mathrm{mph}\). (a) Determine the compass heading that the pilot needs to reach her destination. (b) How long will it take her to reach her destination? Round to the nearest minute.

A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of \(30^{\circ}\) with the ground. Express the force vector \(\mathbf{F}\) in terms of i and \(\mathbf{j}\).

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ y=-3 $$

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j} $$

In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive \(x\) -axis. (a) Write the location of Wrigley Field using rectangular coordinates. (b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees. (c) Guaranteed Rate Field, home of the White \(\operatorname{Sox},\) is located at 35 th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of Guaranteed Rate Field using rectangular coordinates. (d) Write the location of Guaranteed Rate Field using polar coordinates. Use the east direction for the polar axis. Express \(\theta\) in degrees.
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