Liczby Zespolone: Trygonometryczne Dzielenie

Hey guys! Today we're diving deep into the awesome world of complex numbers, specifically tackling a division problem using their trigonometric form. This stuff might seem a bit intimidating at first, but trust me, once you get the hang of it, it's super cool and actually pretty straightforward. We're going to break down this particular problem: (10(cos 126° + i sin 126°)) ÷ (2(cos 72° + i sin 72°)). This is a classic example of how powerful the trigonometric form of complex numbers can be when you need to perform operations like division. So, buckle up, grab your calculators (or just your brains!), and let's get this math party started!

Understanding Complex Numbers in Trigonometric Form

Before we jump into the division, let's quickly recap what complex numbers in trigonometric form are all about. You'll often see them written as $z = r(\cos \theta + i \sin \theta)$, where '$r$' is the modulus (or magnitude) of the complex number, and '$\theta$' is the argument (or angle). Think of '$r$' as how far the number is from the origin on the complex plane, and '$\theta$' as the angle it makes with the positive real axis. This form is incredibly useful because it simplifies multiplication and division, which can get messy with the standard $a + bi$ form. When we multiply complex numbers in trigonometric form, we multiply their moduli and add their arguments. Pretty neat, right? And guess what? Division is just as elegant: we divide the moduli and subtract the arguments. This is the key principle we'll be using today, and it's a total game-changer for simplifying these kinds of operations.

Our problem involves two complex numbers already presented in this glorious trigonometric form. The first one, let's call it $z_1$, is $10(\cos 126° + i \sin 126°)$. Here, the modulus $r_1$ is 10, and the argument $\theta_1$ is 126°. The second complex number, $z_2$, is $2(\cos 72° + i \sin 72°)$, giving us a modulus $r_2$ of 2 and an argument $\theta_2$ of 72°. The task is to compute $z_1 \div z_2$. Remembering the rule for division, we know we need to divide the moduli and subtract the arguments. So, the resulting modulus will be $r_1 / r_2$, and the resulting argument will be $\theta_1 - \theta_2$. This is where the magic happens, transforming a potentially complex fraction into a much simpler expression.

The Division Formula in Action

Alright, let's apply the division rule we just discussed to our specific problem. We have $z_1 = 10(\cos 126° + i \sin 126°)$ and $z_2 = 2(\cos 72° + i \sin 72°)$. To find $z_1 / z_2$, we follow the formula:

$z_1 / z_2 = (r_1 / r_2) (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$

First, let's handle the moduli. We divide $r_1$ by $r_2$: $10 \div 2 = 5$. Easy peasy!

Next, we deal with the arguments. We subtract $\theta_2$ from $\theta_1$: $126° - 72° = 54°$.

Putting it all together, the result of the division is $5(\cos 54° + i \sin 54°)$.

This is the complex number in its trigonometric form after performing the division. It's a single complex number with a modulus of 5 and an argument of 54°. See? Not so scary after all! The trigonometric form really shines when it comes to division. It allows us to perform the operation on the magnitudes and angles separately, making the whole process much more manageable and less prone to errors compared to dealing with the $a+bi$ form directly for division.

Now, the original question also asked us to present the result in trigonometric form, which we've just done! The result is $5(\cos 54° + i \sin 54°)$. This form is often super useful for further calculations, especially if you need to do more multiplications or divisions, or even raise the number to a power (that's where De Moivre's Theorem comes in, which is another cool topic!). So, for this specific problem, we've successfully computed the division and expressed the answer in the required trigonometric form. It's a testament to the elegance and efficiency of using the trigonometric representation of complex numbers for these types of operations. It's like having a secret shortcut that makes complex math feel a little less complex!

Why Trigonometric Form Rocks for Division

Let's chat a bit more about why the trigonometric form is such a superstar for division. Imagine you had to divide $(10 + 0i)$ by $(2 + 0i)$ just as a thought experiment, or something much more complicated like $(a+bi)/(c+di)$. You'd probably have to use the conjugate to rationalize the denominator, which involves a bunch of multiplications and divisions of real numbers. It can get pretty messy, right? Especially if the numbers aren't as simple as in our example.

However, when complex numbers are in the form $z = r(\cos \theta + i \sin \theta)$, division becomes a piece of cake. The formula, as we saw, is $(r_1(\cos \theta_1 + i \sin \theta_1)) \div (r_2(\cos \theta_2 + i \sin \theta_2)) = (r_1/r_2)(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$. This means you just take the ratio of the moduli ($r_1/r_2$) and the difference of the arguments ($\theta_1 - \theta_2$). That's it! No complex algebraic manipulations needed. It's a direct, clean calculation.

This simplicity is invaluable in many areas of math and engineering, particularly in fields like electrical engineering and signal processing, where complex numbers are used extensively to represent quantities like impedance and phase. Being able to quickly divide or multiply these complex quantities using their polar (trigonometric) form saves a ton of time and reduces the chance of calculation errors. It's like having a specialized tool that's perfectly designed for the job, making tasks that would be cumbersome with other methods feel effortless. The geometric interpretation is also quite intuitive: dividing complex numbers geometrically means scaling by the ratio of their lengths and rotating by the difference of their angles. The trigonometric form directly captures these geometric operations.

So, for our problem, $(10(\cos 126° + i \sin 126°)) \div (2(\cos 72° + i \sin 72°))$, the moduli are 10 and 2, and the arguments are 126° and 72°. The division yields a modulus of $10/2 = 5$ and an argument of $126° - 72° = 54°$. Therefore, the result is $5(\cos 54° + i \sin 54°)$. This is the simplest representation in trigonometric form. It beautifully demonstrates the power and elegance of this mathematical tool. It's not just about getting the answer; it's about understanding how and why the trigonometric form makes these operations so much easier. It's a fundamental concept that unlocks a deeper understanding of complex numbers and their applications.

Final Result and What it Means

So, after all that, we've arrived at our final answer. The division of $(10(\cos 126° + i \sin 126°))$ by $(2(\cos 72° + i \sin 72°))$ is $5(\cos 54° + i \sin 54°)$. This is the complex number expressed in its standard trigonometric form. It represents a number on the complex plane that is 5 units away from the origin (its modulus) and is located at an angle of 54 degrees counterclockwise from the positive real axis (its argument).

This result is significant because it consolidates the operation into a single, clear representation. We started with two complex numbers, each with its own magnitude and direction, and through the simple process of dividing magnitudes and subtracting angles, we obtained a new complex number that encapsulates the outcome of that division. The beauty of the trigonometric form is that it preserves this clarity. Even if the angles were more complex or the moduli involved fractions, the process remains the same: divide the $r$'s, subtract the $\theta$'s. This makes it incredibly reliable for calculations.

Think about it: instead of dealing with potentially messy fractions and complex conjugates in the $a+bi$ form, we performed a simple division and subtraction. This is why understanding and utilizing the trigonometric form of complex numbers is so crucial, especially when dealing with multiplication and division. It's a fundamental tool in the mathematician's and engineer's toolkit. We've successfully completed the task, arriving at a clean and precise answer in the required format. This demonstrates a solid grasp of complex number operations in their trigonometric representation. Keep practicing these, guys, and you'll be navigating the complex plane like a pro in no time!

Remember, the key takeaways are: for division of complex numbers in trigonometric form $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, the result is $z_1/z_2 = (r_1/r_2)(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$. Apply this rule, and you're golden. Happy calculating!