Complex Number Subtraction Finding W-z In Rectangular Form

Hey guys! Today, we're diving into the fascinating world of complex numbers. We've got two complex numbers, w and z, presented in their trigonometric (or polar) forms, and our mission, should we choose to accept it, is to find w - z and express the result in rectangular form. Buckle up, because this is going to be a fun ride!

The Problem at Hand

Let's start by stating the problem clearly. We're given:

• w = 2(cos(210°) + isin(210°))
• z = 2(cos(330°) + isin(330°))

Our goal is to calculate w - z and write the answer in the form a + bi, where a and b are real numbers. This a + bi form is what we call the rectangular form of a complex number. To tackle this, we'll need to convert w and z from their trigonometric forms into rectangular forms first. Understanding complex numbers, trigonometric forms, and rectangular forms is crucial here. Complex numbers, which extend the real number system by including the imaginary unit i (where i² = -1), are fundamental in various fields like engineering, physics, and mathematics. Their representation in trigonometric form, r(cos θ + isin θ), provides insights into their magnitude (r) and direction (θ) in the complex plane. Converting to rectangular form (a + bi) allows for straightforward arithmetic operations. The process involves using trigonometric identities to find the real (a) and imaginary (b) components, which are essential for expressing the final result accurately. This conversion not only simplifies the subtraction operation in this problem but also enhances our ability to visualize and manipulate complex numbers in different contexts, solidifying our understanding of their multifaceted nature and applicability.

Converting to Rectangular Form

The key to solving this problem lies in converting the complex numbers w and z from their trigonometric forms to rectangular forms. A complex number in trigonometric form r(cos θ + isin θ) can be converted to rectangular form a + bi using the following relationships:

• a = rcos θ
• b = rsin θ

Converting w

For w = 2(cos(210°) + isin(210°)), we have r = 2 and θ = 210°. Let's find the rectangular form components:

• a = 2cos(210°)
• b = 2sin(210°)

Now, we need to recall our trigonometric values for 210°. Remember that 210° is in the third quadrant, where both cosine and sine are negative. We can think of 210° as 180° + 30°. Using the reference angle of 30°, we know:

• cos(210°) = -cos(30°) = -√3/2
• sin(210°) = -sin(30°) = -1/2

Plugging these values back in, we get:

• a = 2(-√3/2) = -√3
• b = 2(-1/2) = -1

Therefore, w in rectangular form is -√3 - i. The conversion process involves understanding the relationships between trigonometric functions and angles in the complex plane. By breaking down the angle 210° into its reference angle of 30°, we can easily recall the values of cosine and sine. The use of the unit circle and trigonometric identities is essential in determining the signs and magnitudes of these values. The real part, a, is found by multiplying the magnitude (r) by the cosine of the angle, while the imaginary part, b, is found by multiplying the magnitude by the sine of the angle. This conversion is not just a mathematical manipulation; it's a bridge between different representations of complex numbers, allowing us to visualize them geometrically and perform arithmetic operations more efficiently. The rectangular form provides a straightforward way to add, subtract, multiply, and divide complex numbers, making it a fundamental tool in complex number arithmetic. The careful application of trigonometric principles ensures the accuracy of the conversion, laying the groundwork for the subsequent steps in solving the problem.

Converting z

Next, let's convert z = 2(cos(330°) + isin(330°)) to rectangular form. Here, r = 2 and θ = 330°. We calculate:

• a = 2cos(330°)
• b = 2sin(330°)

330° lies in the fourth quadrant, where cosine is positive and sine is negative. We can think of 330° as 360° - 30°. Again, using the 30° reference angle:

• cos(330°) = cos(30°) = √3/2
• sin(330°) = -sin(30°) = -1/2

Substituting these values:

• a = 2(√3/2) = √3
• b = 2(-1/2) = -1

So, z in rectangular form is √3 - i. The process of converting z to rectangular form mirrors that of w, but with key differences due to the angle's position in the fourth quadrant. Understanding the quadrant in which the angle lies is crucial for determining the signs of the trigonometric functions. In this case, 330° is in the fourth quadrant, where cosine is positive and sine is negative. This understanding allows us to correctly apply the reference angle of 30° and find the values of cos(330°) and sin(330°). The real part, a, is calculated as 2 * (√3/2) = √3, and the imaginary part, b, is calculated as 2 * (-1/2) = -1. These calculations demonstrate the importance of attention to detail and a solid grasp of trigonometric principles. The accurate conversion of z to its rectangular form is essential for the subsequent subtraction operation. This conversion underscores the versatility of trigonometric forms in representing complex numbers and highlights the practicality of converting between trigonometric and rectangular forms to facilitate different types of mathematical operations. The resulting rectangular form, √3 - i, is a crucial step in finding the final solution to the problem.

Finding w - z

Now that we have w = -√3 - i and z = √3 - i in rectangular form, we can easily find w - z. Remember, to subtract complex numbers in rectangular form, we subtract the real parts and subtract the imaginary parts:

• w - z = (-√3 - i) - (√3 - i)

Distribute the negative sign:

• w - z = -√3 - i - √3 + i

Combine like terms:

• w - z = (-√3 - √3) + (-i + i)
• w - z = -2√3 + 0i

Therefore, w - z = -2√3. The subtraction of complex numbers in rectangular form is a straightforward process, but it's essential to maintain accuracy by carefully distributing the negative sign and combining like terms. The real parts, -√3 and √3, are subtracted to give -2√3, while the imaginary parts, -i and -(-i), cancel each other out, resulting in 0i. This step highlights the simplicity of performing arithmetic operations on complex numbers in rectangular form. The result, -2√3, is a real number, which indicates that the imaginary components of w and z canceled out during the subtraction. This outcome underscores the importance of understanding the properties of complex numbers and how they interact under different operations. The final result, -2√3, is the rectangular form of the difference between w and z, and it provides a concise and clear answer to the problem. This process not only demonstrates the practical application of complex number arithmetic but also reinforces the fundamental principles of algebraic manipulation.

The Final Answer

So, guys, we've successfully navigated through the world of complex numbers! We converted w and z from their trigonometric forms to rectangular forms, and then we subtracted them to find w - z. Our final answer, expressed in rectangular form, is:

• w - z = -2√3*

This is equivalent to -2√3 + 0i, but we typically omit the 0i term. This journey through complex numbers illustrates the power of converting between different forms to simplify calculations and gain deeper insights. The process of converting complex numbers from trigonometric to rectangular form and then performing subtraction highlights the flexibility and utility of complex number arithmetic. By breaking down the problem into manageable steps, we were able to apply trigonometric principles and algebraic manipulations to arrive at the final solution. The result, -2√3, is a real number, which may seem surprising given the initial forms of w and z. However, this outcome underscores the importance of performing the calculations to fully understand the relationships between complex numbers. This exercise not only reinforces the fundamental concepts of complex number arithmetic but also demonstrates the practical application of these concepts in solving mathematical problems. The final answer, presented clearly and concisely, serves as a testament to our ability to navigate the complex world of mathematics with confidence and precision.

I hope this explanation was helpful and clear. Keep exploring the fascinating world of mathematics!