Finding The Conjugate Of A Complex Number Quotient Using Trigonometric Form

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Introduction

In the realm of complex numbers, exploring their properties and operations often leads to fascinating mathematical insights. This article delves into a specific problem involving the division of two complex numbers, $z_1 = 1 + \sqrt{3}i$ and $z_2 = -1 + i$, and the subsequent determination of the conjugate of their quotient. To tackle this, we'll leverage the elegance and power of trigonometric forms, providing a comprehensive, step-by-step guide that's accessible to both students and enthusiasts alike. Our journey will not only solve the given problem but also illuminate the broader concepts and techniques involved in complex number manipulation.

Our main goal is to find $\overline{\left(\frac{z_1}{z_2}\right)}$ by employing the trigonometric representations of $z_1$ and $z_2$. This involves several key steps: converting the complex numbers into their trigonometric forms, performing the division in trigonometric form, and finally, finding the conjugate of the resulting complex number. Each of these steps will be thoroughly explained and justified, providing a clear understanding of the underlying principles.

Before we dive into the solution, let's briefly recap some essential concepts. A complex number can be expressed in the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$). The conjugate of $z$, denoted by $\overline{z}$, is obtained by changing the sign of the imaginary part, i.e., $\overline{z} = a - bi$. The trigonometric form (also known as the polar form) of a complex number expresses it in terms of its magnitude (or modulus) $r$ and argument (or angle) $\theta$, as $z = r(\cos\theta + i\sin\theta)$. Understanding these basics is crucial for navigating the intricacies of complex number operations.

This article is structured to provide a clear and coherent explanation of the solution process. We'll begin by converting $z_1$ and $z_2$ into their trigonometric forms, carefully detailing the steps involved in finding the modulus and argument of each number. Next, we'll perform the division operation in trigonometric form, highlighting the rules and simplifications that apply. Finally, we'll determine the conjugate of the resulting quotient, solidifying our understanding of this important concept. By the end of this article, you'll have a firm grasp of how to manipulate complex numbers using trigonometric forms and how to find the conjugate of a quotient, empowering you to tackle similar problems with confidence.

Converting $z_1$ and $z_2$ into Trigonometric Form

The cornerstone of our approach lies in expressing the complex numbers $z_1 = 1 + \sqrt{3}i$ and $z_2 = -1 + i$ in their trigonometric forms. The trigonometric form of a complex number, as we mentioned earlier, provides a powerful way to visualize and manipulate complex numbers. To convert a complex number $z = a + bi$ into trigonometric form, we need to determine its modulus ($r$) and argument ($\theta$).

The modulus of a complex number $z = a + bi$, denoted by $|z|$, represents its distance from the origin in the complex plane and is calculated as $r = |z| = \sqrt{a^2 + b^2}$. The argument of $z$, denoted by $arg(z)$, is the angle $\theta$ formed between the positive real axis and the line segment connecting the origin to the point representing $z$ in the complex plane. It's crucial to consider the quadrant in which the complex number lies to determine the correct argument. The argument can be found using the formula $\theta = \arctan(\frac{b}{a})$, but we must adjust the angle based on the quadrant.

Let's start with $z_1 = 1 + \sqrt{3}i$. The real part is $a = 1$, and the imaginary part is $b = \sqrt{3}$. The modulus of $z_1$ is:

r1=∣z1∣=12+(3)2=1+3=4=2r_1 = |z_1| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2

To find the argument $\theta_1$, we use the formula:

θ1=arctan⁡(31)=arctan⁡(3)\theta_1 = \arctan(\frac{\sqrt{3}}{1}) = \arctan(\sqrt{3})

Since $z_1$ lies in the first quadrant (both real and imaginary parts are positive), the argument is simply the principal value of the arctangent, which is $\frac{\pi}{3}$ radians or 60 degrees. Therefore, the trigonometric form of $z_1$ is:

z1=2(cos⁡(π3)+isin⁡(π3))z_1 = 2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))

Now, let's move on to $z_2 = -1 + i$. Here, the real part is $a = -1$, and the imaginary part is $b = 1$. The modulus of $z_2$ is:

r2=∣z2∣=(−1)2+12=1+1=2r_2 = |z_2| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}

For the argument $\theta_2$, we have:

θ2=arctan⁡(1−1)=arctan⁡(−1)\theta_2 = \arctan(\frac{1}{-1}) = \arctan(-1)

Since $z_2$ lies in the second quadrant (real part is negative, and the imaginary part is positive), we need to adjust the arctangent value. The principal value of $\arctan(-1)$ is $-\frac{\pi}{4}$, which corresponds to the fourth quadrant. To find the argument in the second quadrant, we add $\pi$ to this value:

θ2=−π4+π=3π4\theta_2 = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}

Thus, the trigonometric form of $z_2$ is:

z2=2(cos⁡(3π4)+isin⁡(3π4))z_2 = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))

Having successfully converted both $z_1$ and $z_2$ into their trigonometric forms, we're now well-equipped to proceed with the division operation. The trigonometric form not only simplifies the division process but also provides a clear geometric interpretation of the operation. The modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the difference of the arguments. This elegant property makes the trigonometric form a powerful tool for complex number arithmetic.

Dividing $z_1$ by $z_2$ in Trigonometric Form

With $z_1$ and $z_2$ expressed in their trigonometric forms, the division operation becomes remarkably straightforward. Recall that we have:

z1=2(cos⁡(π3)+isin⁡(π3))z_1 = 2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))

z2=2(cos⁡(3π4)+isin⁡(3π4))z_2 = \sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))

The division of two complex numbers in trigonometric form follows a specific rule: 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:

z1z2=r1r2(cos⁡(θ1−θ2)+isin⁡(θ1−θ2))\frac{z_1}{z_2} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))

This rule elegantly encapsulates the geometric interpretation of complex number division: the modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the difference of the arguments. This simplification makes the trigonometric form incredibly useful for performing complex number arithmetic.

Applying this rule to our problem, we have:

z1z2=22(cos⁡(π3−3π4)+isin⁡(π3−3π4))\frac{z_1}{z_2} = \frac{2}{\sqrt{2}}(\cos(\frac{\pi}{3} - \frac{3\pi}{4}) + i\sin(\frac{\pi}{3} - \frac{3\pi}{4}))

First, let's simplify the modulus:

22=222=2\frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}

Next, we simplify the argument:

π3−3π4=4π−9π12=−5π12\frac{\pi}{3} - \frac{3\pi}{4} = \frac{4\pi - 9\pi}{12} = \frac{-5\pi}{12}

Therefore, the quotient $\frac{z_1}{z_2}$ in trigonometric form is:

z1z2=2(cos⁡(−5π12)+isin⁡(−5π12))\frac{z_1}{z_2} = \sqrt{2}(\cos(\frac{-5\pi}{12}) + i\sin(\frac{-5\pi}{12}))

This expression represents the complex number resulting from the division of $z_1$ by $z_2$ in its trigonometric form. The modulus of the quotient is $\sqrt{2}$, and the argument is $-\frac{5\pi}{12}$. This negative argument indicates that the complex number lies in the fourth quadrant of the complex plane.

Now that we have successfully divided $z_1$ by $z_2$, our final step is to find the conjugate of this quotient. The conjugate of a complex number in trigonometric form is obtained by simply changing the sign of the imaginary part, which corresponds to negating the argument. This property makes finding the conjugate in trigonometric form remarkably simple and highlights the elegance of this representation.

Finding the Conjugate of the Quotient

The final piece of our puzzle is to determine the conjugate of the quotient $\frac{z_1}{z_2}$. We've already established that:

z1z2=2(cos⁡(−5π12)+isin⁡(−5π12))\frac{z_1}{z_2} = \sqrt{2}(\cos(\frac{-5\pi}{12}) + i\sin(\frac{-5\pi}{12}))

Recall that the conjugate of a complex number $z = a + bi$, denoted by $\overline{z}$, is obtained by changing the sign of the imaginary part, resulting in $\overline{z} = a - bi$. In the trigonometric form, if $z = r(\cos\theta + i\sin\theta)$, then its conjugate is given by:

z‾=r(cos⁡θ−isin⁡θ)\overline{z} = r(\cos\theta - i\sin\theta)

Alternatively, using the properties of cosine and sine, we can write:

z‾=r(cos⁡(−θ)+isin⁡(−θ))\overline{z} = r(\cos(-\theta) + i\sin(-\theta))

This elegant result shows that to find the conjugate in trigonometric form, we simply negate the argument while keeping the modulus the same. This is a direct consequence of the symmetry of the cosine function and the antisymmetry of the sine function. Geometrically, the conjugate of a complex number is its reflection across the real axis in the complex plane.

Applying this to our quotient, we have:

(z1z2)‾=2(cos⁡(−5π12)−isin⁡(−5π12))\overline{\left(\frac{z_1}{z_2}\right)} = \sqrt{2}(\cos(\frac{-5\pi}{12}) - i\sin(\frac{-5\pi}{12}))

Using the property that $\cos(-\theta) = \cos(\theta)$ and $\sin(-\theta) = -\sin(\theta)$, we can rewrite this as:

(z1z2)‾=2(cos⁡(5π12)+isin⁡(5π12))\overline{\left(\frac{z_1}{z_2}\right)} = \sqrt{2}(\cos(\frac{5\pi}{12}) + i\sin(\frac{5\pi}{12}))

This is the trigonometric form of the conjugate of the quotient $\frac{z_1}{z_2}$. The modulus remains $\sqrt{2}$, but the argument is now $ rac{5\pi}{12}$, which is the negation of the original argument. This confirms our understanding of how conjugation affects the argument of a complex number.

We have successfully determined the conjugate of the quotient $\frac{z_1}{z_2}$ using the trigonometric forms of $z_1$ and $z_2$. This process involved converting the complex numbers into trigonometric form, performing the division operation using the trigonometric form division rule, and finally, finding the conjugate by negating the argument. This comprehensive approach showcases the power and elegance of trigonometric forms in handling complex number operations.

Conclusion

In this article, we embarked on a journey to find the conjugate of the quotient of two complex numbers, $z_1 = 1 + \sqrt{3}i$ and $z_2 = -1 + i$, using the trigonometric form. We meticulously walked through the steps, starting with the conversion of $z_1$ and $z_2$ into their trigonometric representations. This involved calculating the modulus and argument for each complex number, paying careful attention to the quadrant in which they lie.

Next, we performed the division of $z_1$ by $z_2$ in trigonometric form, utilizing the rule that the modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the difference of the arguments. This step highlighted the efficiency and clarity that the trigonometric form brings to complex number division.

Finally, we determined the conjugate of the resulting quotient. We demonstrated that finding the conjugate in trigonometric form simply involves negating the argument, while the modulus remains unchanged. This elegant property further underscores the utility of the trigonometric form in complex number manipulations.

Our solution culminated in the following expression for the conjugate of the quotient:

(z1z2)‾=2(cos⁡(5π12)+isin⁡(5π12))\overline{\left(\frac{z_1}{z_2}\right)} = \sqrt{2}(\cos(\frac{5\pi}{12}) + i\sin(\frac{5\pi}{12}))

This result showcases the power of the trigonometric form in simplifying complex number operations and providing a clear geometric interpretation. The modulus of the conjugate is $\sqrt{2}$, and the argument is $\frac{5\pi}{12}$, reflecting the geometric relationship between a complex number and its conjugate.

This exploration not only provides a solution to the specific problem at hand but also reinforces the fundamental concepts of complex numbers, their trigonometric representations, and the operations that can be performed on them. The techniques and insights gained from this analysis can be readily applied to a wide range of complex number problems, empowering students and enthusiasts to delve deeper into this fascinating area of mathematics. The trigonometric form is a valuable tool in the complex number arsenal, offering a visual and computationally efficient way to tackle various problems. By mastering these concepts, one can unlock a deeper understanding of complex numbers and their applications in various fields, such as physics, engineering, and computer science.