Trigonometric Form Of Complex Number Z = (1 + I) / I

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Hey guys! Ever wondered how to express a complex number in its trigonometric form? It's actually a pretty cool way to represent these numbers, and in this article, we're going to break it down step-by-step. We'll tackle the specific example of z = (1 + i) / i. So, buckle up and let's dive into the world of complex numbers!

Understanding Complex Numbers

Before we jump into the trigonometric form, let's quickly recap what complex numbers are. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1). The a part is called the real part, and the b part is called the imaginary part.

Think of complex numbers as living in a 2D plane, called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. So, a complex number a + bi can be plotted as a point (a, b) in this plane. This visual representation is super helpful in understanding the trigonometric form.

Now, why do we even bother with complex numbers? Well, they pop up in all sorts of fields, from electrical engineering to quantum mechanics. They allow us to solve equations that have no solutions in the real number system and provide a powerful tool for analyzing oscillations and waves. So, understanding them is definitely worth the effort!

Why Trigonometric Form Matters

You might be thinking, "Okay, I get complex numbers, but why do we need a trigonometric form?" Great question! The trigonometric form offers several advantages, especially when it comes to certain operations like multiplication and division. It also provides a geometric interpretation that can be really insightful.

In essence, the trigonometric form expresses a complex number in terms of its magnitude (or absolute value) and its angle (or argument) with respect to the positive real axis in the complex plane. This is analogous to using polar coordinates instead of Cartesian coordinates in the regular 2D plane. Instead of describing a point by its x and y coordinates, we describe it by its distance from the origin and the angle it makes with the x-axis.

This representation makes certain operations much easier to visualize and perform. For instance, multiplying two complex numbers in trigonometric form simply involves multiplying their magnitudes and adding their angles. Division is equally straightforward: divide the magnitudes and subtract the angles. This is way simpler than the algebraic manipulations required when dealing with the a + bi form.

Moreover, the trigonometric form connects complex numbers to trigonometry, allowing us to use trigonometric identities and functions to solve complex number problems. It's like having another tool in your toolbox, ready to tackle a wider range of challenges.

Step-by-Step Solution for z = (1 + i) / i

Alright, let's get our hands dirty and find the trigonometric form of z = (1 + i) / i. We'll break it down into manageable steps so you can follow along easily.

Step 1: Simplify the Complex Number

First things first, we need to simplify the given complex number. We have z = (1 + i) / i. To get rid of the i in the denominator, we'll multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of i is -i. Remember, the complex conjugate of a complex number a + bi is a - bi.

So, let's multiply:

z = (1 + i) / i * (-i / -i) = (-i - i²) / (-i²)

Now, recall that i² = -1. Substitute that in:

z = (-i - (-1)) / (-(-1)) = (-i + 1) / 1 = 1 - i

Great! We've simplified the complex number to z = 1 - i. This form is much easier to work with.

Step 2: Find the Modulus (Magnitude)

The modulus (or magnitude) of a complex number z = a + bi, denoted as |z|, is its distance from the origin in the complex plane. It's calculated using the Pythagorean theorem:

|z| = √(a² + b²)

In our case, z = 1 - i, so a = 1 and b = -1. Let's plug these values into the formula:

|z| = √(1² + (-1)²) = √(1 + 1) = √2

So, the modulus of z is √2. This tells us that the distance of the point representing z from the origin in the complex plane is √2 units.

Step 3: Find the Argument (Angle)

The argument of a complex number z = a + bi, denoted as arg(z) or θ, is the angle it makes with the positive real axis in the complex plane. We can find the argument using the following formula:

θ = arctan(b / a)

However, we need to be careful about the quadrant in which the complex number lies, as the arctan function only gives angles in the range (-π/2, π/2). In our case, z = 1 - i, so a = 1 and b = -1. Let's calculate the initial angle:

θ' = arctan(-1 / 1) = arctan(-1) = -π/4

Since a = 1 and b = -1, the complex number z lies in the fourth quadrant. The angle we calculated, -Ï€/4, is indeed in the fourth quadrant, so it's the correct argument.

Therefore, the argument of z is -Ï€/4 (or -45 degrees).

Step 4: Express in Trigonometric Form

Now that we have the modulus and the argument, we can write the complex number in its trigonometric form. The trigonometric form of a complex number z is:

z = |z|(cos θ + i sin θ)

We found that |z| = √2 and θ = -π/4. Let's substitute these values:

z = √2(cos(-π/4) + i sin(-π/4))

And that's it! We've successfully expressed the complex number z = (1 + i) / i in its trigonometric form.

Final Answer

The trigonometric form of the complex number z = (1 + i) / i is:

z = √2(cos(-π/4) + i sin(-π/4))

Or, equivalently:

z = √2(cos(7π/4) + i sin(7π/4))

(Since -Ï€/4 and 7Ï€/4 represent the same angle)

Key Takeaways

Let's quickly recap the key steps we took to solve this problem:

  1. Simplify the complex number: We multiplied the numerator and denominator by the complex conjugate to get rid of the i in the denominator.
  2. Find the modulus: We used the formula |z| = √(a² + b²) to calculate the magnitude of the complex number.
  3. Find the argument: We used the arctan function, being careful to consider the quadrant in which the complex number lies.
  4. Express in trigonometric form: We plugged the modulus and argument into the formula z = |z|(cos θ + i sin θ).

Understanding these steps will help you convert any complex number into its trigonometric form. Remember, this form is particularly useful for multiplication, division, and visualizing complex number operations.

Practice Makes Perfect

Now that you've seen how to convert a complex number to trigonometric form, it's time to practice! Try converting some other complex numbers, such as 1 + i, -2 + 2i, or -3 - 4i. The more you practice, the more comfortable you'll become with the process.

You can also explore the geometric interpretation of complex number operations in the complex plane. For example, try plotting the complex numbers and their trigonometric forms to see how multiplication corresponds to rotation and scaling.

Conclusion

So, there you have it! We've successfully found the trigonometric form of the complex number z = (1 + i) / i. Hopefully, this step-by-step guide has made the process clear and understandable. Remember, complex numbers might seem a bit intimidating at first, but with practice and a good understanding of the underlying concepts, you'll be able to tackle them with confidence. Keep exploring, keep learning, and keep having fun with math!