Decoding Complex Transformations Quotient Of Complex Numbers Z And W On The Complex Plane

Jul 23, 2025 by Admin 90 views

$Understanding Transformations on the Complex Plane The quotient of complex numbers on the complex plane reveals fascinating geometric transformations. Let's dive into understanding how dividing complex numbers affects their representation on this plane. In this article, we will explore the transformation resulting from the quotient of two complex numbers, $z=8(\cos(30^\circ) + i \sin(30^\circ))$ and $w=2(\cos(5^\circ) + i \sin(5^\circ))$. We will dissect the mathematical mechanics involved, provide a step-by-step breakdown of the solution, and discuss the underlying concepts to ensure a comprehensive understanding. This includes using trigonometric forms of complex numbers, understanding their geometric interpretations, and leveraging properties of division in polar form. ## Complex Numbers in Polar Form Before we delve into the quotient, let's refresh the concept of the polar form of complex numbers. **Representing complex numbers** in polar form provides a geometric interpretation that greatly simplifies operations like multiplication and division. A complex number $z = a + bi$ can be expressed in polar form as: $z = r(\cos(\theta) + i \sin(\theta))$ where: - $r$ is the magnitude (or modulus) of $z$, calculated as $r = \sqrt{a^2 + b^2}$. - $\theta$ is the argument of $z$, which is the angle formed with the positive real axis, found using $ \theta = \arctan(\frac{b}{a})$. The expression $(\cos(\theta) + i \sin(\theta))$ is often abbreviated as $cis(\theta)$. So, $z$ can be written as $z = r \cdot cis(\theta)$. **Polar form** elegantly captures both the magnitude and direction of a complex number, making it a powerful tool for visualizing complex number operations. ## Diving into the Given Complex Numbers We are given two complex numbers: $z = 8(\cos(30^\circ) + i \sin(30^\circ))$ $w = 2(\cos(5^\circ) + i \sin(5^\circ))$ These are already in polar form, which makes our task significantly easier. Here: - For $z$, the magnitude $r_1 = 8$ and the argument $\theta_1 = 30^\circ$. - For $w$, the magnitude $r_2 = 2$ and the argument $\theta_2 = 5^\circ$. Understanding these components is crucial because when we divide complex numbers in polar form, we divide their magnitudes and subtract their arguments. ## The Division of Complex Numbers in Polar Form **Dividing complex numbers** in polar form follows a specific rule that simplifies the process: If $z_1 = r_1(\cos(\theta_1) + i \sin(\theta_1))$ and $z_2 = r_2(\cos(\theta_2) + i \sin(\theta_2))$, then: $\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$ This rule stems from the properties of trigonometric functions and complex number multiplication. To really get what's happening here, it's essential to consider the geometric interpretation. Think of complex numbers as vectors in the complex plane. **Dividing by a complex number** is akin to scaling and rotating the original vector. The scaling factor is determined by the ratio of the magnitudes, and the rotation is determined by the difference in the arguments. ## Calculating the Quotient $z/w$ Now, let's apply this rule to find the quotient of our given complex numbers: $z = 8(\cos(30^\circ) + i \sin(30^\circ))$ $w = 2(\cos(5^\circ) + i \sin(5^\circ))$ Using the division rule, we have: $\frac{z}{w} = \frac{8}{2} [\cos(30^\circ - 5^\circ) + i \sin(30^\circ - 5^\circ)]$ Simplify the magnitudes and arguments: $\frac{z}{w} = 4 [\cos(25^\circ) + i \sin(25^\circ)]$ So, the quotient $z/w$ is a complex number with a magnitude of 4 and an argument of $25^\circ$. This means that the complex number $z$ is transformed by scaling its magnitude by a factor of 4 and rotating it counterclockwise by $25^\circ$ when divided by $w$. ## Geometric Interpretation of the Transformation The geometric interpretation is where the magic happens. **Visualizing complex numbers** on the complex plane helps us understand the effect of division. $z = 8(\cos(30^\circ) + i \sin(30^\circ))$ represents a vector with a length of 8, making an angle of $30^\circ$ with the positive real axis. $w = 2(\cos(5^\circ) + i \sin(5^\circ))$ represents a vector with a length of 2, making an angle of $5^\circ$ with the positive real axis. When we divide $z$ by $w$, we obtain: $\frac{z}{w} = 4(\cos(25^\circ) + i \sin(25^\circ))$ This new complex number represents a vector with a length of 4 (8 divided by 2) and an angle of $25^\circ$ (30 degrees minus 5 degrees). **In essence,** the division has scaled down the magnitude and rotated the vector counterclockwise. The magnitude was reduced because the magnitude of the divisor ($w$) was greater than 1, and the rotation occurred because the argument of $w$ was subtracted from the argument of $z$. ## Describing the Transformation Now, let's articulate the transformation in plain English. **The transformation** of $z$ on the complex plane that gives the quotient of $z$ and $w$ involves two key components: 1. **Scaling:** The magnitude of $z$ is scaled by a factor of $\frac{8}{2} = 4$. This means that the distance from the origin to the point representing the complex number is multiplied by 4. 2. **Rotation:** The complex number $z$ is rotated counterclockwise by an angle of $30^\circ - 5^\circ = 25^\circ$. This is because dividing by a complex number effectively subtracts its argument from the argument of the dividend. **Therefore,** the transformation can be described as a scaling by a factor of 4 and a rotation of $25^\circ$ counterclockwise on the complex plane. ## Summary of the Solution To recap, here’s a concise summary of how we arrived at the description of the transformation: 1. We identified the **polar forms** of the complex numbers $z$ and $w$. 2. We applied the **division rule** for complex numbers in polar form: $\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$. 3. We calculated the **quotient** $\frac{z}{w} = 4(\cos(25^\circ) + i \sin(25^\circ))$. 4. We interpreted the **geometric meaning** of the result, recognizing it as a scaling and a rotation on the complex plane. 5. We described the **transformation** as a scaling by a factor of 4 and a rotation of $25^\circ$ counterclockwise. ## Conclusion Understanding transformations on the complex plane is a cornerstone of complex number theory. **Dividing complex numbers** in polar form not only simplifies calculations but also provides a clear geometric interpretation. The process involves scaling the magnitude and rotating the complex number, offering insights into how complex numbers interact. In our specific example, dividing $z$ by $w$ resulted in a scaling by a factor of 4 and a rotation of $25^\circ$ counterclockwise. This methodical breakdown enhances our understanding and appreciation of complex number operations. By mastering these concepts, you'll be well-equipped to tackle more complex problems in mathematics and related fields. # Decoding Complex Transformations Understanding the Transformation of z on the Complex Plane ## Repair Input Keyword What transformation on the complex plane represents the quotient of $z=8(\\cos(30^\circ)+i \\sin(30^\circ))$ and $w=2(\\cos(5^\circ)+i \\sin(5^\circ))$? ## SEO Title Decoding Complex Transformations Quotient of Complex Numbers z and w on the Complex Plane$