Problem 39 - Precalculus

Question

Use a calculator to express each complex number in polar form. $$4.02-2.11 i$$

Step-by-Step Solution

Answer

The polar form is approximately \(4.54 \text{cis}(-27.97^\circ)\).

Step 1: Understand the Task

We need to convert the given complex number, which is in rectangular form, to polar form. The complex number given is: \(4.02 - 2.11i\).

Step 2: Calculate the Magnitude

The magnitude \( r \) of a complex number \( a + bi \) is calculated using the formula \( r = \sqrt{a^2 + b^2} \). For the complex number \(4.02 - 2.11i\), we calculate:\[ r = \sqrt{(4.02)^2 + (-2.11)^2} \]

Step 3: Simplify the Magnitude Expression

Compute the square inside the root:\[ r = \sqrt{16.1604 + 4.4521} = \sqrt{20.6125} \]

Step 4: Compute the Magnitude

Use a calculator to solve:\[ r \approx \sqrt{20.6125} \approx 4.54 \]

Step 5: Calculate the Angle (Argument)

The angle \( \theta \) is calculated using the formula \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \). For this complex number, \( a = 4.02 \) and \( b = -2.11 \):\[ \theta = \tan^{-1} \left( \frac{-2.11}{4.02} \right) \]

Step 6: Compute the Angle using a Calculator

Calculate the tangent:\[ \theta \approx \tan^{-1} \left( \frac{-2.11}{4.02} \right) \approx \tan^{-1} (-0.5249) \]The result is \( \theta \approx -27.97^\circ \).

Step 7: Express in Polar Form

Using the values of \( r \) and \( \theta \), the polar form of the complex number is:\[ 4.54 \text{cis}(-27.97^\circ) \]

Key Concepts

Magnitude of Complex Number

The magnitude of a complex number is essential when converting it to polar form. Think of the magnitude as the distance from the origin to the point in the complex plane. For a complex number given by \(a + bi\), you can find its magnitude with the equation:\
\[ r = \sqrt{a^2 + b^2} \]
This equation is similar to the Pythagorean theorem, which helps determine the hypotenuse of a right triangle. In our problem, the complex number is \(4.02 - 2.11i\). To find its magnitude, substitute \(a = 4.02\) and \(b = -2.11\) into the equation. You'll calculate:\
\[ r = \sqrt{(4.02)^2 + (-2.11)^2} = \sqrt{20.6125} \]
Using a calculator, you find that \( r \approx 4.54 \). This magnitude tells us how far the number is from the center of the complex plane.

Argument of Complex Number

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. This angle is crucial for expressing the number in polar form. To calculate the argument, \( \theta \), use the formula:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
For the complex number \(4.02 - 2.11i\), \(a = 4.02\) and \(b = -2.11\). Applying these values, we find:
\[ \theta = \tan^{-1} \left( \frac{-2.11}{4.02} \right) \]
This gives a result of approximately \(-27.97^\circ\).
The negative angle indicates the point is below the real axis in the complex plane, which is typical for complex numbers with a negative imaginary component. In polar coordinates, this direction information is just as vital as the magnitude, as it specifies the number's exact position in the plane.

Rectangular to Polar Conversion

Converting a complex number from rectangular form \((a + bi)\) to polar form involves using both its magnitude and argument. Polar form is useful for understanding the size and direction of complex numbers as it consists of a magnitude and an angle. The polar form is given by:
\[ r \text{cis}(\theta) \]
where \( \text{cis} \) is just a shorthand for \( \cos\theta + i\sin\theta \).
For the given complex number \(4.02 - 2.11i\), we already found that the magnitude \(r\) is approximately \(4.54\) and the argument \(\theta\) is approximately \(-27.97^\circ\).
Thus, the polar form of this complex number can be written as:
\[ 4.54 \text{cis}(-27.97^\circ) \]
This form showcases both the distance from the origin and the direction in which the number lies, providing a clear picture of its status in the complex plane. This form is especially handy in multiplying and dividing complex numbers, making it a powerful tool in computations.