If |z| = 4 And Arg Z = 5π/6, What Is The Value Of Z?

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In the realm of mathematics, complex numbers hold a significant position, extending the concept of real numbers to include an imaginary unit, typically denoted as i, where i² = -1. Complex numbers are expressed in the form z = a + bi, where a and b are real numbers, and a represents the real part (Re(z)) and b represents the imaginary part (Im(z)). To delve deeper into complex numbers, we need to understand two crucial concepts: the modulus (|z|) and the argument (arg z). The modulus of a complex number represents its distance from the origin in the complex plane, while the argument signifies the angle formed between the positive real axis and the line connecting the complex number to the origin. Understanding these concepts is essential for solving problems involving complex numbers. In this article, we will explore a specific problem involving a complex number z with a given modulus and argument, and we will determine its rectangular form.

Problem Statement: Decoding the Complex Number z

Let's consider a complex number z where the modulus, denoted as |z|, is equal to 4, and the argument, denoted as arg z, is equal to 5π/6. The objective is to determine the value of z in its rectangular form, which is a + bi, where a and b are real numbers. This problem provides a good opportunity to apply our understanding of modulus and argument to find the real and imaginary parts of a complex number. The given information essentially provides us with the magnitude and direction of the complex number in the complex plane. We need to use this information to find the coordinates of the complex number in the complex plane, which will give us the real and imaginary parts. This involves using trigonometric relationships to connect the modulus and argument to the real and imaginary parts of the complex number. Understanding this connection is crucial for working with complex numbers in various applications, including electrical engineering, quantum mechanics, and signal processing.

The Modulus and Argument Unveiled

The modulus of a complex number z = a + bi, denoted as |z|, is the distance from the origin (0, 0) to the point (a, b) in the complex plane. It is calculated using the Pythagorean theorem: |z| = √(a² + b²). In simpler terms, the modulus gives us the magnitude or the absolute value of the complex number. The argument of a complex number z, denoted as arg z, is the angle θ formed between the positive real axis and the line connecting the origin to the point (a, b) in the complex plane. The argument is typically measured in radians and can be found using trigonometric functions. Specifically, we have tan θ = b/ a. However, it's crucial to consider the quadrant in which the complex number lies to determine the correct angle. The argument is not unique; adding multiples of 2π to the argument will result in the same complex number. Therefore, we often consider the principal argument, which lies in the interval (-π, π] or [0, 2π). Understanding the modulus and argument is essential because they provide a way to represent complex numbers in polar form, which is particularly useful for certain operations such as multiplication and division.

Connecting the Dots: From Polar to Rectangular Form

To express a complex number z in rectangular form (a + bi) given its modulus |z| and argument arg z = θ, we employ the following relationships:

  • a = |z| cos θ
  • b = |z| sin θ

These equations stem from basic trigonometry in the complex plane. The real part a is the projection of the complex number onto the real axis, which is given by the modulus multiplied by the cosine of the argument. Similarly, the imaginary part b is the projection of the complex number onto the imaginary axis, which is given by the modulus multiplied by the sine of the argument. These relationships are fundamental in converting between polar and rectangular forms of complex numbers. The polar form, which uses the modulus and argument, is particularly useful for understanding the magnitude and direction of the complex number, while the rectangular form, which uses the real and imaginary parts, is useful for performing addition and subtraction. The ability to convert between these forms is a crucial skill in complex number arithmetic and analysis. This conversion allows us to leverage the advantages of each form depending on the specific problem we are trying to solve.

Solution: Unraveling the Complex Number

Given |z| = 4 and arg z = 5π/6, we can find the real and imaginary parts of z using the formulas mentioned above.

  1. Calculate the real part (a):

    a = |z| cos(arg z) = 4 cos(5π/6)

    Since cos(5π/6) = -√3/2,

    a = 4 * (-√3/2) = -2√3

    The real part of the complex number is found by multiplying the modulus by the cosine of the argument. The cosine of 5π/6 is -√3/2, and multiplying this by the modulus 4 gives us -2√3. This calculation is a direct application of the relationship between polar and rectangular coordinates. The result indicates that the real component of the complex number is negative, which aligns with the fact that the argument 5π/6 lies in the second quadrant of the complex plane, where the real part is negative and the imaginary part is positive. Understanding the behavior of trigonometric functions in different quadrants is essential for accurately converting between polar and rectangular forms.

  2. Calculate the imaginary part (b):

    b = |z| sin(arg z) = 4 sin(5π/6)

    Since sin(5π/6) = 1/2,

    b = 4 * (1/2) = 2

    The imaginary part is calculated similarly, but using the sine function instead of the cosine function. The sine of 5π/6 is 1/2, and multiplying this by the modulus 4 gives us 2. This positive value for the imaginary part confirms that the complex number lies in the second quadrant. The imaginary part represents the vertical component of the complex number in the complex plane, and its positive value indicates that the complex number is above the real axis. These calculations highlight the importance of knowing the values of trigonometric functions for common angles, as they are frequently used in complex number problems.

Therefore, the complex number z in rectangular form is:

z = a + bi = -2√3 + 2i

The Final Answer: Decoding the Correct Option

Comparing our solution with the given options, we find that the correct answer is:

  1. -2√3 + 2i

This process of converting from polar form (modulus and argument) to rectangular form (real and imaginary parts) is a fundamental skill in complex number manipulation. The ability to accurately perform this conversion is crucial for solving a wide range of problems in mathematics, physics, and engineering. Understanding the geometric interpretation of complex numbers in the complex plane further enhances this skill, allowing for a more intuitive understanding of the relationships between the modulus, argument, real part, and imaginary part. This problem serves as a good example of how these concepts are applied in practice.

Conclusion: Mastering Complex Numbers

This problem illustrates the fundamental relationship between the modulus, argument, and rectangular form of a complex number. By understanding these concepts and applying trigonometric principles, we can effectively convert between polar and rectangular forms. This skill is essential for solving a wide array of problems in complex analysis and its applications. The key takeaway is that the modulus and argument provide a geometric way to represent complex numbers, while the rectangular form provides an algebraic way. The ability to seamlessly transition between these representations is crucial for mastering complex numbers. Furthermore, this problem reinforces the importance of understanding trigonometric functions and their values for common angles. Practice with similar problems will solidify your understanding and build your proficiency in working with complex numbers. As you delve deeper into complex numbers, you will encounter more complex operations and applications, but the fundamental concepts discussed here will remain essential. Mastering these concepts will not only help you in academic settings but also in various real-world applications where complex numbers play a vital role.