Complex Number To Rectangular Form: Z_1 = 7, Θ_1 = 270°

Hey guys! Today, we're diving into the fascinating world of complex numbers and tackling the problem of converting a complex number from its polar form to its rectangular form. Specifically, we're going to express the complex number z₁, which has a magnitude of 7 and an angle of 270 degrees, in the form a + bi, where a and b are real numbers. This is a fundamental concept in complex number theory, and mastering it will open doors to more advanced topics. Understanding complex numbers is crucial in many fields, including engineering, physics, and computer science. So, let's jump right in and break down this conversion step-by-step. We will explore the polar and rectangular forms of complex numbers, the conversion formulas, and then apply them to solve our specific problem. By the end of this guide, you'll have a solid grasp of how to convert complex numbers between these two forms, empowering you to tackle similar problems with confidence. Remember, the key to understanding math is practice, so feel free to try out more examples after this one! Let's get started and unravel the mystery of complex number conversions together!

Understanding Polar and Rectangular Forms

Before we dive into the conversion, let's briefly review the two forms of representing complex numbers: polar and rectangular. The rectangular form expresses a complex number as z = a + bi, where a is the real part and b is the imaginary part. Think of this like coordinates on a graph, where a is the x-coordinate and b is the y-coordinate. On the other hand, the polar form represents a complex number using its magnitude (or modulus), denoted as |z|, and its angle (or argument), denoted as θ. The magnitude |z| represents the distance of the complex number from the origin in the complex plane, while the angle θ represents the angle formed between the positive real axis and the line connecting the origin to the complex number. The polar form is often written as z = |z|(cos θ + i sin θ). This form is particularly useful when dealing with rotations and multiplications of complex numbers. The connection between these two forms is crucial for manipulating complex numbers effectively. The real part, a, can be found using the formula a = |z| cos θ, and the imaginary part, b, can be found using the formula b = |z| sin θ. Understanding this relationship allows us to move seamlessly between the polar and rectangular representations. So, with this basic understanding in place, we can now tackle the specific problem of converting z₁ from polar to rectangular form. We will use these formulas as the backbone of our conversion process. Remember, the polar form gives us a visual representation of the complex number's distance and direction, while the rectangular form gives us the real and imaginary components. Let's see how these concepts come together in our example!

Conversion Formulas: The Key to Unlocking the Rectangular Form

The heart of converting from polar to rectangular form lies in two simple yet powerful formulas. These formulas bridge the gap between the magnitude and angle representation of a complex number (polar form) and its real and imaginary components (rectangular form). They are derived directly from trigonometric relationships within the complex number plane. The first formula allows us to calculate the real part, a, of the complex number: a = |z| cos θ. This formula essentially projects the magnitude |z| onto the real axis using the cosine of the angle θ. The second formula allows us to calculate the imaginary part, b, of the complex number: b = |z| sin θ. This formula, similarly, projects the magnitude |z| onto the imaginary axis using the sine of the angle θ. By calculating a and b using these formulas, we can then express the complex number in its rectangular form, z = a + bi. These formulas are not just mathematical tools; they are the key to understanding the geometric relationship between the polar and rectangular representations. Think of them as translators, allowing us to convert between the language of magnitude and angle to the language of real and imaginary components. Mastery of these formulas is essential for working with complex numbers in various applications. They are the foundation for performing operations like addition, subtraction, multiplication, and division in both forms. So, make sure you understand these formulas inside and out. Now, let's put these formulas into action and convert our specific complex number, z₁, into its rectangular form. We will carefully apply each formula, step by step, to arrive at the solution. Are you ready to see the magic of these formulas unfold?

Applying the Formulas to z₁ = 7(cos 270° + i sin 270°)

Now, let's get our hands dirty and apply these conversion formulas to our specific problem. We're given that z₁ has a magnitude |z₁| = 7 and an angle θ₁ = 270°. Our goal is to find the real part, a, and the imaginary part, b, so we can express z₁ in the rectangular form z₁ = a + bi. Let's start by finding the real part, a. Using the formula a = |z₁| cos θ₁, we substitute the given values: a = 7 cos 270°. Now, we need to recall the value of cos 270°. If you visualize the unit circle, 270° corresponds to the point at the bottom of the circle, where the x-coordinate (which represents cosine) is 0. Therefore, cos 270° = 0. Plugging this into our equation, we get a = 7 * 0 = 0. So, the real part of z₁ is 0. Next, let's find the imaginary part, b. Using the formula b = |z₁| sin θ₁, we substitute the given values: b = 7 sin 270°. Again, we need to recall the value of sin 270°. From the unit circle, the y-coordinate at 270° (which represents sine) is -1. Therefore, sin 270° = -1. Plugging this into our equation, we get b = 7 * (-1) = -7. So, the imaginary part of z₁ is -7. We've now successfully calculated both the real and imaginary parts of z₁. This means we're ready to express z₁ in rectangular form. Let's put it all together!

Expressing z₁ in Rectangular Form: The Final Result

Alright guys, we've done the hard work of calculating the real and imaginary parts of z₁. We found that the real part, a, is 0, and the imaginary part, b, is -7. Now, the final step is to express z₁ in the rectangular form z₁ = a + bi. Simply substitute the values we found for a and b: z₁ = 0 + (-7)i. We can simplify this to z₁ = -7i. And there you have it! We've successfully converted the complex number z₁ from its polar form with magnitude 7 and angle 270° to its rectangular form, which is z₁ = -7i. This result makes sense geometrically. A complex number with a magnitude of 7 and an angle of 270° lies on the negative imaginary axis, 7 units away from the origin. This corresponds perfectly to the rectangular form z₁ = -7i, where there is no real component and a negative imaginary component of -7. This example beautifully illustrates how the polar and rectangular forms are just two different ways of representing the same complex number. They provide complementary perspectives, and knowing how to convert between them is a powerful tool in complex number analysis. So, are you feeling confident about converting complex numbers now? Remember, practice makes perfect! Try converting other complex numbers from polar to rectangular form to solidify your understanding. You can even try going the other way, from rectangular to polar form! Keep exploring the fascinating world of complex numbers, and you'll be amazed at the power and beauty of this mathematical concept.