Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

-0.03472 + 0.19696i

Step 1 - 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, \(r = 0.2\), \(\theta = \frac{5 \pi}{9}\).

Step 2 - 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.

Step 3 - Calculate \(\cos \frac{5 \pi}{9}\)

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

Step 4 - Calculate \(\sin \frac{5 \pi}{9}\)

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

Step 5 - Substitute back into the rectangular coordinates

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

Step 6 - 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 $$-0.03472 + 0.19696i.$$

Key Concepts

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.