Problem 17 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-12-5 i $$

Step-by-Step Solution

Answer

The polar form of \( z = -12 - 5i \) is \( 13 (\cos(3.605) + i\sin(3.605)) \); \( \operatorname{Re}(z) = -12 \), \( \operatorname{Im}(z) = -5 \), \( |z| = 13 \), \( \arg(z) \approx 3.605 \) radians.

Step 1: Identify the Real and Imaginary Parts

The complex number given is \( z = -12 - 5i \). Here, the real part of \( z \), \( \operatorname{Re}(z) \), is the coefficient of the non-imaginary part, which is \(-12\). The imaginary part of \( z \), \( \operatorname{Im}(z) \), is the coefficient of the imaginary unit \( i \), which is \(-5\).

Step 2: Calculate the Magnitude

The magnitude \( |z| \) of a complex number \( z = a + bi \) is given by the equation \( |z| = \sqrt{a^2 + b^2} \). Thus, for \( z = -12 - 5i \), we have:\[|z| = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.\]

Step 3: Find the Argument

The argument \( \arg(z) \) of the complex number is the angle formed with the positive x-axis. Calculate it using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). For \( z = -12 - 5i \) (both \( a \) and \( b \) are negative), the angle is in the third quadrant:\[\theta = \tan^{-1}\left(\frac{-5}{-12}\right) = \tan^{-1}\left(\frac{5}{12}\right)\]This gives a positive angle when calculated directly, but since we are in the third quadrant, we adjust this by adding \( \pi \), so:\[\arg(z) = \pi + \tan^{-1}\left(\frac{5}{12}\right)\]

Step 4: Express in Polar Form

To express \( z \) in polar form, use \( z = |z|\left(\cos(\arg z) + i\sin(\arg z)\right) \). Thus, the polar form of \( z = -12 - 5i \) is:\[z = 13\left(\cos(\arg z) + i\sin(\arg z)\right) \approx 13\left(\cos(3.605) + i\sin(3.605)\right)\]

Step 5: Conclusion

Summarizing the findings:- \( \operatorname{Re}(z) = -12 \)- \( \operatorname{Im}(z) = -5 \)- \( |z| = 13 \)- \( \arg(z) = \pi + \tan^{-1}\left(\frac{5}{12}\right) \approx 3.605 \) radians- The polar representation of \( z \) is approximately \( 13 (\cos(3.605) + i\sin(3.605)) \).

Key Concepts

Magnitude of complex numbers

To get started with the polar representation of a complex number, a key part is understanding how to find its magnitude. The magnitude of a complex number, also referred to as its modulus, indicates how "big" the number is. If you imagine a complex number on a two-dimensional plane, the magnitude is the distance between this number and the origin. This helps in understanding how far the number is from the center of the plane.
To calculate the magnitude of a complex number given in the form \( z = a + bi \), you use the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

Here, \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively. In our example \( z = -12 - 5i \), substituting \( a = -12 \) and \( b = -5 \) gives:

• \( |z| = \sqrt{(-12)^2 + (-5)^2} = \sqrt{169} = 13 \)

This result means our complex number is 13 units away from the origin.

Argument of complex numbers

Another important aspect of converting complex numbers to polar form is finding the argument of the complex number. The argument of a complex number is the angle formed between the positive x-axis and the line representing the complex number.
For a complex number \( z = a + bi \), the argument can directionally locate \( z \) in the complex plane and is computed using the arctangent function:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

However, when \( a \) or \( b \) is negative, you must consider the quadrant in which the complex number lies. Our example \( z = -12 - 5i \) lies in the third quadrant where both \( a \) and \( b \) are negative. Therefore:

• \( \arg(z) = \pi + \tan^{-1}\left(\frac{5}{12}\right) \approx 3.605 \) radians

This adjustment ensures that the argument reflects the correct direction of \( z \) in the complex plane.

Polar form conversion

After finding both the magnitude and the argument, the next step is converting the complex number to polar form. The polar form expresses the complex number using radius and angle instead of real and imaginary parts. It's useful for visualizing, calculating powers, and finding the roots of complex numbers.
The polar representation of a complex number is given by:

• \( z = |z|(\cos(\arg z) + i\sin(\arg z)) \)

This form utilizes the magnitude \( |z| \) and the argument \( \arg(z) \) which we calculated earlier. Applying these to \( z = -12 - 5i \), it transforms into polar form as:

• \( z = 13(\cos(3.605) + i\sin(3.605)) \)

This transformation reveals \( z \) not by coordinates, but through its position and distance in the complex plane, opening the door to more advanced operations.