Complex Numbers: Trigonometric Form & Operations

Hey guys! Let's dive into the fascinating world of complex numbers, specifically focusing on converting them into trigonometric form and tackling some operations. This stuff might seem a bit abstract at first, but trust me, it's super useful in many areas of math and engineering. We'll break it down step by step, so you'll be a complex number whiz in no time!

Converting Complex Numbers to Trigonometric Form

First off, let's talk about converting complex numbers to trigonometric form. Why do we even bother doing this? Well, the trigonometric form makes certain operations, like multiplication and division, much easier to handle. A complex number, typically written in the form a + bi, can be represented in trigonometric form as r(cos θ + i sin θ). Here, r is the magnitude (or modulus) of the complex number, and θ is the argument (or angle) it makes with the positive real axis. To kick things off, let's remember the core concepts. The conversion to trigonometric form relies heavily on understanding the complex plane, which is basically a 2D graph where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number. The trigonometric form provides an alternative representation using polar coordinates, which can be incredibly handy for certain operations and visualizations. Now, let's tackle the process in detail. The first step is to calculate the modulus, r, which represents the distance from the origin to the point representing the complex number in the complex plane. This is found using the Pythagorean theorem: r = √(a² + b²), where a is the real part and b is the imaginary part of the complex number. This is a crucial step, as it gives us the magnitude of the complex number, which is a key component in the trigonometric form. Next up is finding the argument, θ, which is the angle the complex number makes with the positive real axis. This can be found using the arctangent function: θ = arctan(b/a). However, a crucial caveat here is that the arctangent function only gives values in the range (-π/2, π/2), so you might need to adjust the angle based on the quadrant in which the complex number lies. This is because the arctangent function only considers the ratio of the imaginary and real parts, not their signs, which are crucial for determining the correct quadrant. So, if the complex number is in the second or third quadrant, you'll need to add π to the result to get the correct angle. Once we have both r and θ, we can plug them into the trigonometric form: z = r(cos θ + i sin θ). This form expresses the complex number in terms of its magnitude and direction, making it easier to visualize and manipulate. The trigonometric form is especially useful when multiplying or dividing complex numbers, as these operations become much simpler when the numbers are expressed in this form. For instance, multiplying two complex numbers in trigonometric form simply involves multiplying their magnitudes and adding their arguments. Similarly, dividing two complex numbers involves dividing their magnitudes and subtracting their arguments. This is a significant advantage over performing these operations in the standard rectangular form, which can be more cumbersome. Now that we have the general framework down, let's apply these steps to the specific complex numbers given in the problem. Each complex number presents its own unique challenge, whether it's dealing with fractions, negative signs, or purely real numbers. By working through these examples, we will solidify our understanding and gain confidence in our ability to convert any complex number to trigonometric form. So, let's roll up our sleeves and get started on these conversions!

1. (5, 5)

Let's start with the complex number (5, 5). This is equivalent to 5 + 5i. To convert this complex number into trigonometric form, we first need to find r, the magnitude. Using the formula r = √(a² + b²), we have:

r = √(5² + 5²) = √(25 + 25) = √50 = 5√2

Next, we find the argument θ. Since both the real and imaginary parts are positive, the complex number lies in the first quadrant. We can use the arctangent function:

θ = arctan(5/5) = arctan(1) = 45° or π/4 radians

Therefore, the trigonometric form of (5, 5) is:

5√2(cos(π/4) + i sin(π/4))

This complex number lies in the first quadrant, and we have successfully converted it to trigonometric form. Guys, remember to always double-check the quadrant to make sure your angle is correct!

2. (-3/2, -5/2)

Now, let's tackle the complex number (-3/2, -5/2), which is equivalent to -3/2 - 5/2i. Again, we start by finding the magnitude, r:

r = √((-3/2)² + (-5/2)²) = √(9/4 + 25/4) = √(34/4) = √(34)/2

Next, we need to determine the argument θ. Since both the real and imaginary parts are negative, this complex number lies in the third quadrant. Using the arctangent function:

θ' = arctan((-5/2) / (-3/2)) = arctan(5/3)

This gives us an angle in the first quadrant. To find the correct angle in the third quadrant, we need to add π radians (180°) to this value:

θ = arctan(5/3) + π

Let's keep the angle in this form for now, as it's the exact representation. So, the trigonometric form of (-3/2, -5/2) is:

(√(34)/2)(cos(arctan(5/3) + π) + i sin(arctan(5/3) + π))

See how we handled the negative signs and made sure we got the correct quadrant? It's crucial to pay attention to these details. You've got this!

3. 8

This one's a bit simpler. The complex number 8 can be written as 8 + 0i. To find the magnitude, r:

r = √(8² + 0²) = √64 = 8

Now, let's find the argument, θ. Since the imaginary part is zero and the real part is positive, this number lies on the positive real axis. Therefore:

θ = 0 radians or 0°

So, the trigonometric form of 8 is:

8(cos(0) + i sin(0))

This is a straightforward example, but it's good to see how real numbers fit into the complex number framework.

4. (-5, 5)

The complex number (-5, 5) is equivalent to -5 + 5i. Let's find the magnitude, r:

r = √((-5)² + 5²) = √(25 + 25) = √50 = 5√2

Now, let's find the argument, θ. Since the real part is negative and the imaginary part is positive, this complex number lies in the second quadrant. Using the arctangent function:

θ' = arctan(5 / -5) = arctan(-1) = -π/4

This gives us an angle in the fourth quadrant (or a negative angle). To get the angle in the second quadrant, we need to add π radians (180°) to this value:

θ = -π/4 + π = 3π/4

Therefore, the trigonometric form of (-5, 5) is:

5√2(cos(3π/4) + i sin(3π/4))

We're making progress! Notice how the quadrant dictates the adjustment we make to the arctangent result.

5. 12 + 16i

For the complex number 12 + 16i, let's find the magnitude, r:

r = √(12² + 16²) = √(144 + 256) = √400 = 20

Next, we find the argument, θ. Since both parts are positive, this is in the first quadrant:

θ = arctan(16/12) = arctan(4/3)

We'll leave the angle as arctan(4/3) for an exact representation. The trigonometric form is:

20(cos(arctan(4/3)) + i sin(arctan(4/3)))

6. 15

Similar to the case with 8, the complex number 15 is 15 + 0i. The magnitude, r, is:

r = √(15² + 0²) = √225 = 15

The argument, θ, is 0 radians (0°) because it lies on the positive real axis. So, the trigonometric form is:

15(cos(0) + i sin(0))

Determining Results of Operations with Complex Numbers

Okay, now that we've mastered converting complex numbers to trigonometric form, let's move on to performing operations with them. Remember, the trigonometric form really shines when it comes to multiplication and division. The problem gives us Z₁ = (cos 45° + i sin 45°). It looks like the question is incomplete, because it mentions Z₂ and Z₃ but doesn't provide them. Therefore, I can't fully answer question 7 without knowing what Z₂ and Z₃ are, and what operations need to be performed. But, let's discuss the general principles and how we'd approach it if we had the other numbers.

Let's say we had Z₂ = r₂(cos θ₂ + i sin θ₂) and we wanted to find Z₁ * Z₂. Here's the magic: Multiplying complex numbers in trigonometric form is super straightforward. You multiply the magnitudes and add the angles:

Z₁ * Z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

So, if we had a Z₂, we'd just multiply the magnitudes (1 in this case, since Z₁ has a magnitude of 1) and add the angles (45° + θ₂). Similarly, for division, you divide the magnitudes and subtract the angles:

Z₁ / Z₂ = (r₁/r₂)[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]

If we were given operations like addition or subtraction, it's generally easier to convert the complex numbers back to their rectangular form (a + bi) first, perform the operation, and then convert back to trigonometric form if needed. Remember, cos θ + i sin θ can be converted back to rectangular form using the values of cosine and sine for the angle θ. For example, cos 45° = √2/2 and sin 45° = √2/2, so Z₁ in rectangular form is (√2/2 + i√2/2). Without the complete information about Z₂ and Z₃ and the desired operations, I can't give a specific numerical answer for question 7, but these are the general principles you'd apply.

Conclusion

So there you have it, guys! We've covered converting complex numbers to trigonometric form and discussed how to perform operations with them. Remember, finding the magnitude and argument are key to the conversion process, and watching out for the correct quadrant is crucial. The trigonometric form makes multiplication and division a breeze, while addition and subtraction are often easier in rectangular form. Keep practicing, and you'll become a pro at complex number manipulations! If you have any more questions or want to dive deeper into specific operations, just let me know!