Product Of Complex Numbers In Polar Form A Step-by-Step Solution

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In the realm of mathematics, complex numbers hold a special allure, extending the familiar number line into a two-dimensional plane. These numbers, composed of a real and an imaginary part, open doors to a fascinating world of mathematical operations and applications. Among these operations, multiplication stands out as a fundamental concept, particularly when dealing with complex numbers expressed in polar form. This article delves into the intricacies of multiplying complex numbers in polar form, providing a comprehensive guide to understanding and mastering this essential mathematical skill.

We embark on a journey to unravel the product of two complex numbers given in their polar representations: 2(extcos(45∘)+iextsin(45∘))2( ext{cos}(45^{\circ}) + i ext{sin}(45^{\circ})) and 5(extcos(30∘)+iextsin(30∘))5( ext{cos}(30^{\circ}) + i ext{sin}(30^{\circ})). By the end of this exploration, you will not only be able to compute the product but also gain a deeper appreciation for the underlying principles that govern complex number multiplication. So, let's dive in and unravel the beauty of complex number multiplication!

Understanding Complex Numbers in Polar Form

Before we delve into the multiplication process, it's crucial to grasp the concept of polar form in the context of complex numbers. A complex number, typically written as z=a+biz = a + bi, where 'a' represents the real part and 'b' represents the imaginary part, can also be expressed in polar form. This alternative representation offers a geometric perspective, describing the complex number in terms of its magnitude (or modulus) and its angle (or argument).

The polar form of a complex number is given by z=r(extcos(θ)+iextsin(θ))z = r( ext{cos}(\theta) + i ext{sin}(\theta)), where:

  • rr represents the magnitude (or modulus) of the complex number, which is the distance from the origin to the point representing the complex number in the complex plane. It is calculated as r=a2+b2r = \sqrt{a^2 + b^2}.
  • θ\theta represents the angle (or argument) of the complex number, which is the angle formed between the positive real axis and the line segment connecting the origin to the point representing the complex number in the complex plane. It is calculated as θ=extarctan(ba)\theta = ext{arctan}(\frac{b}{a}).

In essence, the polar form provides a way to visualize complex numbers as points in a plane, where 'r' determines the distance from the origin and 'θ\theta' determines the direction. This representation proves particularly useful when performing operations like multiplication and division, as it simplifies the calculations and offers a more intuitive understanding of the results.

The Significance of Polar Form in Multiplication

The polar form of complex numbers shines when it comes to multiplication. It transforms the process into a remarkably elegant and straightforward operation. The key lies in the following property: when multiplying two complex numbers in polar form, we multiply their magnitudes and add their angles.

Mathematically, if we have two complex numbers:

z1=r1(cos(θ1)+isin(θ1))z_1 = r_1(\text{cos}(\theta_1) + i\text{sin}(\theta_1))

and

z2=r2(cos(θ2)+isin(θ2))z_2 = r_2(\text{cos}(\theta_2) + i\text{sin}(\theta_2))

Then their product, z1∗z2z_1 * z_2, is given by:

z1∗z2=r1r2(cos(θ1+θ2)+isin(θ1+θ2))z_1 * z_2 = r_1r_2(\text{cos}(\theta_1 + \theta_2) + i\text{sin}(\theta_1 + \theta_2))

This property stems from the trigonometric identities for the cosine and sine of a sum of angles. It elegantly encapsulates the geometric interpretation of complex number multiplication: the magnitude of the product is the product of the magnitudes, and the angle of the product is the sum of the angles.

This transformation from multiplication and addition in rectangular form to multiplication of magnitudes and addition of arguments in polar form significantly simplifies calculations, especially when dealing with multiple complex numbers or complex powers.

Step-by-Step Solution: Multiplying the Given Complex Numbers

Now, let's apply our understanding of polar form multiplication to the specific problem at hand. We are tasked with finding the product of the complex numbers:

z1=2(cos(45∘)+isin(45∘))z_1 = 2(\text{cos}(45^{\circ}) + i\text{sin}(45^{\circ}))

and

z2=5(cos(30∘)+isin(30∘))z_2 = 5(\text{cos}(30^{\circ}) + i\text{sin}(30^{\circ}))

Step 1: Identify the Magnitudes and Angles

The first step involves identifying the magnitudes and angles of the two complex numbers. From the given expressions, we can readily see that:

  • For z1z_1, the magnitude r1=2r_1 = 2 and the angle θ1=45∘\theta_1 = 45^{\circ}.
  • For z2z_2, the magnitude r2=5r_2 = 5 and the angle θ2=30∘\theta_2 = 30^{\circ}.

These values directly correspond to the 'r' and 'θ\theta' components in the general polar form representation.

Step 2: Multiply the Magnitudes

Next, we multiply the magnitudes of the two complex numbers:

r1∗r2=2∗5=10r_1 * r_2 = 2 * 5 = 10

This result, 10, represents the magnitude of the product complex number.

Step 3: Add the Angles

Now, we add the angles of the two complex numbers:

θ1+θ2=45∘+30∘=75∘\theta_1 + \theta_2 = 45^{\circ} + 30^{\circ} = 75^{\circ}

This result, 75∘^{\circ}, represents the angle of the product complex number.

Step 4: Construct the Product in Polar Form

Finally, we combine the results from steps 2 and 3 to construct the product complex number in polar form. Using the property that the product's magnitude is the product of the magnitudes and the product's angle is the sum of the angles, we obtain:

z1∗z2=10(cos(75∘)+isin(75∘))z_1 * z_2 = 10(\text{cos}(75^{\circ}) + i\text{sin}(75^{\circ}))

The Final Answer

Therefore, the product of 2(cos(45∘)+isin(45∘))2(\text{cos}(45^{\circ}) + i\text{sin}(45^{\circ})) and 5(cos(30∘)+isin(30∘))5(\text{cos}(30^{\circ}) + i\text{sin}(30^{\circ})) is:

10(cos(75∘)+isin(75∘))10(\text{cos}(75^{\circ}) + i\text{sin}(75^{\circ}))

Thus, the answer to the question is: The product is 10\boxed{10} (cos(75∘)+isin(75∘))(\text{cos}(\boxed{75}^{\circ}) + i\text{sin}(\boxed{75}^{\circ})).

Visualizing the Multiplication

To further solidify your understanding, let's visualize this multiplication in the complex plane. Each complex number can be represented as a vector originating from the origin. The magnitude corresponds to the length of the vector, and the angle corresponds to the angle the vector makes with the positive real axis.

When we multiply two complex numbers, we are essentially scaling and rotating one vector by the other. The magnitude of the product vector is the product of the magnitudes of the original vectors, and the angle of the product vector is the sum of the angles of the original vectors.

Imagine z1z_1 as a vector with length 2 and angle 45∘^{\circ}, and z2z_2 as a vector with length 5 and angle 30∘^{\circ}. Multiplying them results in a vector with length 10 (2 * 5) and angle 75∘^{\circ} (45∘^{\circ} + 30∘^{\circ}). This geometric interpretation provides a visual and intuitive understanding of complex number multiplication.

Converting Back to Rectangular Form (Optional)

While the polar form is convenient for multiplication, you might sometimes need to convert the result back to rectangular form (a + bi). To do this, we simply evaluate the cosine and sine functions and distribute the magnitude:

10(cos(75∘)+isin(75∘))=10cos(75∘)+10isin(75∘)10(\text{cos}(75^{\circ}) + i\text{sin}(75^{\circ})) = 10\text{cos}(75^{\circ}) + 10i\text{sin}(75^{\circ})

Using trigonometric identities or a calculator, we can find the approximate values of cos(75∘)\text{cos}(75^{\circ}) and sin(75∘)\text{sin}(75^{\circ}):

  • cos(75∘)≈0.2588\text{cos}(75^{\circ}) \approx 0.2588
  • sin(75∘)≈0.9659\text{sin}(75^{\circ}) \approx 0.9659

Substituting these values, we get:

10(cos(75∘)+isin(75∘))≈10(0.2588)+10i(0.9659)≈2.588+9.659i10(\text{cos}(75^{\circ}) + i\text{sin}(75^{\circ})) \approx 10(0.2588) + 10i(0.9659) \approx 2.588 + 9.659i

Thus, the rectangular form of the product is approximately 2.588 + 9.659i. This step demonstrates the flexibility of complex numbers, allowing us to switch between polar and rectangular forms as needed.

Conclusion

In this article, we embarked on a comprehensive exploration of multiplying complex numbers in polar form. We uncovered the elegance and efficiency of this method, which hinges on the fundamental property of multiplying magnitudes and adding angles. By understanding the geometric interpretation and following the step-by-step solution, you can confidently tackle complex number multiplication problems.

The beauty of polar form lies in its ability to transform a seemingly intricate operation into a straightforward process. This understanding not only enhances your mathematical skills but also provides a deeper appreciation for the rich and interconnected nature of mathematical concepts. As you continue your mathematical journey, remember the power of polar form in simplifying complex number operations and unlocking new mathematical insights.

Multiply Complex Numbers in Polar Form: A Comprehensive Guide