Unlocking Complex Numbers: Trigonometric Form Explained

Hey there, math enthusiasts! Let's dive into the fascinating world of complex numbers and explore how to express them in their trigonometric form. This is super helpful for understanding these numbers and performing operations on them. We'll break down the question, "Given the complex number 2/(1+i), what is its trigonometric form?" step by step, making it easy to understand. So, grab your calculators and let's get started, guys!

Understanding Complex Numbers and Their Forms

Before we jump into the problem, let's refresh our knowledge of complex numbers. 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. The a is the real part, and b is the imaginary part. Complex numbers are used widely in many fields, including electrical engineering, quantum mechanics, and signal processing.

There are several ways to represent a complex number. The most common is the rectangular form (also known as the algebraic form), which is simply a + bi. But another very useful form is the trigonometric form (also known as the polar form). The trigonometric form expresses a complex number using its magnitude (or modulus) and its angle (or argument).

The trigonometric form is written as z = r(cos θ + i sin θ), where:

• z is the complex number.
• r is the magnitude (also called the modulus) of z. It's the distance of the complex number from the origin in the complex plane and is calculated as r = √(a² + b²).
• θ is the argument (or angle) of z. It's the angle between the positive real axis and the line connecting the origin to the point representing the complex number in the complex plane, measured counterclockwise. It can be calculated using trigonometric functions, such as θ = arctan(b/a). However, you need to be careful about the quadrant in which the complex number lies to determine the correct angle.

Using the trigonometric form can make complex number operations, like multiplication and division, much easier. So, understanding how to convert a complex number to its trigonometric form is super important. We will break down the process step by step to solve the problem and show you how it works.

Solving the Problem: Finding the Trigonometric Form of 2/(1+i)

Okay, let's tackle the question: "Given the complex number 2/(1+i), what is its trigonometric form?" Here’s how we'll solve it, breaking it down into manageable steps. First, we need to simplify the given complex number and rewrite it in rectangular form. Then, we will find the magnitude and argument, finally, we can write the number in trigonometric form.

Step 1: Simplify the Complex Number

Our initial complex number is 2/(1+i). To express this in rectangular form (a + bi), we need to eliminate the imaginary part from the denominator. We do this by multiplying both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of (1 + i) is (1 - i). So, let's go ahead and do that:

• Multiply the numerator and denominator by the conjugate:
  • 2/(1+i) * (1-i)/(1-i)
• Multiply out the numerator and the denominator separately.
  • Numerator: 2 * (1 - i) = 2 - 2i
  • Denominator: (1 + i) * (1 - i) = 1 - i² . Since i² = -1, this simplifies to 1 - (-1) = 2
• Simplify the fraction:
  • (2 - 2i) / 2 = 1 - i

So, the simplified complex number in rectangular form is 1 - i. This is very important. Now we have a = 1 and b = -1.

Step 2: Calculate the Magnitude (r)

The magnitude r of a complex number a + bi is calculated using the formula r = √(a² + b²). With our simplified complex number 1 - i, we have a = 1 and b = -1. Let's plug these values into the formula:

• r = √(1² + (-1)²) = √(1 + 1) = √2

So, the magnitude r of the complex number is √2.

Step 3: Calculate the Argument (θ)

The argument θ is the angle that the complex number makes with the positive real axis. We can find this angle using the formula θ = arctan(b/a). Remember, it's also important to check the quadrant to ensure the correct angle is determined.

For our complex number 1 - i, a = 1 and b = -1. So:

• θ = arctan(-1/1) = arctan(-1)

The arctangent of -1 is -π/4 radians (or -45 degrees). However, we need to consider the quadrant. The complex number 1 - i lies in the fourth quadrant (where the real part is positive and the imaginary part is negative). In the complex plane, an angle of -π/4 is indeed in the fourth quadrant. However, we usually express angles between 0 and 2π. Thus, we can also express the angle as θ = 2π - π/4 = 7π/4.

Step 4: Write the Trigonometric Form

Now that we have the magnitude r = √2 and the argument θ = 7π/4, we can write the complex number in its trigonometric form using the formula z = r(cos θ + i sin θ):

• z = √2 (cos(7π/4) + i sin(7π/4)) or z = √2 (cos(-π/4) + i sin(-π/4)).

This is the trigonometric form of the complex number 2/(1+i).

Choosing the Correct Answer from the Options

Now, let’s match our solution with the options provided. The general form is z = r(cos θ + i sin θ). We found that r = √2 and θ = 7π/4 (or -π/4).

Let’s analyze the options given in the question and see which one matches our findings:

• Option a: z = √2 (cos(π/4) + i sin(π/4)) This is incorrect because the angle is wrong.
• Option b: z = √2 (cos(-π/4) + i sin(π/4)) This is incorrect, as the sign for the angle in the sin function should also be negative or the argument should be 7π/4. The cosine is an even function, so cos(-π/4) = cos(π/4).
• Option c: z = √2 (cos(7π/4) + i sin(7π/4)) This is correct, as it matches our calculated magnitude and argument. Remember that 7π/4 is in the fourth quadrant, just like -π/4.
• Option d: z = 0 (cos(π/4) + i sin(π/4)) This is incorrect because the magnitude is 0, which is wrong. The result shows that the magnitude is √2.

Therefore, the correct answer is option c. The trigonometric form of 2/(1+i) is z = √2 (cos(7π/4) + i sin(7π/4)).

Conclusion: Mastering the Trigonometric Form

And there you have it, guys! We've successfully transformed the complex number 2/(1+i) into its trigonometric form. This exercise highlights the importance of understanding the different forms of complex numbers and how to convert between them. Being able to do this opens up a whole new world of possibilities when working with complex numbers. You’ll be able to perform operations like multiplication and division more easily, which can be super useful in solving various mathematical and engineering problems.

Remember, the key steps involve simplifying to rectangular form, calculating the magnitude, finding the argument, and then writing the number in z = r(cos θ + i sin θ) form. Practice these steps with different complex numbers. The more you practice, the easier it will become. Keep practicing, and you'll become a pro at working with complex numbers in no time. If you got any questions, feel free to ask! Happy calculating!