Problem 94 - Precalculus

Question

Another way of using a graphing calculator to represent complex numbers in rectangular form is to enter the real and imaginary parts as a list of two numbers and use the \([\text { sUM }]\) command to find the modulus. Write \(-\sqrt{21}+10 i\) in polar form using the \([\mathrm{SUM}]\) command to find its modulus, and round the angle to the nearest degree.

Step-by-Step Solution

Answer

The polar form is \( 11(\cos 154\degree + i\sin 154\degree) \).

Step 1: Understand Rectangular to Polar Form Conversion

Rectangular form of a complex number is given by \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. For the given complex number \(-\sqrt{21} + 10i\), \( a = -\sqrt{21} \) and \( b = 10 \). In polar form, it's expressed as \( r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

Step 2: Calculate the Modulus Using a Graphing Calculator

The modulus \( r \) is calculated as the square root of the sum of the squares of the real and imaginary parts: \( r = \sqrt{a^2 + b^2} \). Enter the numbers as a list: \([-\sqrt{21}, 10]\), then use the \([\text{SUM}]\) command with their squares to find: \[ r = \sqrt{(-\sqrt{21})^2 + (10)^2} = \sqrt{21 + 100} = \sqrt{121} = 11 \].

Step 3: Calculate the Argument (Angle)

The angle \( \theta \) is found using the tangent function: \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Thus, \( \theta = \tan^{-1}\left(\frac{10}{-\sqrt{21}}\right) \). When you compute this using the calculator, ensure it's in the correct quadrant given \( a < 0 \) and \( b > 0 \), so it lies in the second quadrant.

Step 4: Adjust Angle to Degrees and Quadrant

Compute \( \theta \) in degrees using a calculator: \( \theta \approx \tan^{-1}(-\frac{10}{\sqrt{21}}) + 180\degree \). Simplifying gives \( \theta \approx 154\degree \).

Step 5: Write Polar Form

Now write the polar form using \( r \) and \( \theta \): \( 11(\cos 154\degree + i\sin 154\degree) \).

Key Concepts

Complex Numbers

A complex number is a special type of number that includes a real part and an imaginary part. It is expressed as:

• Real Part: The real number that appears without any imaginary unit (i).
• Imaginary Part: The real number that appears with the imaginary unit \( i \), where \( i^2 = -1 \).

In general, a complex number is written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. It's a handy way to capture both a real and an imaginary component in a single mathematical entity. These numbers are crucial in fields of engineering, physics, and applied mathematics, allowing intricate numbers to be handled with ease. Complex numbers can be graphed on a coordinate plane, with the x-axis representing the real part and the y-axis the imaginary part.

Rectangular Form

The rectangular form is just another name for the standard representation of complex numbers as \( a + bi \). Here, \( a \) and \( b \) are like the coordinates on a plane:

• \( a \) runs along the x-axis, representing the real part.
• \( b \) runs along the y-axis, representing the imaginary part.

Graphically, you can imagine plotting \( a + bi \) similarly to how you plot any point in a 2D coordinate system. This representation helps when performing addition or subtraction since these operations are straightforward in this form. You just add or subtract the real parts and the imaginary parts.

Modulus

The modulus of a complex number is akin to its 'magnitude' or 'length'. It's the distance from the point \( (a, b) \) on the coordinate plane to the origin \( (0, 0) \). To find the modulus of a complex number \( a + bi \):

• Square the real part \( a \)
• Square the imaginary part \( b \)
• Add these squares together.
• Take the square root of this sum to get the modulus \( r \).

Formulaically, it's shown as: \[ r = \sqrt{a^2 + b^2}\]In the context of the exercise, calculations show that the complex number \(-\sqrt{21} + 10i\) has a modulus of 11. This length is an essential piece when converting to polar form.

Argument

The argument of a complex number is the angle formed with the positive real axis. It provides direction, telling you where the number lies relative to the axis. To find the argument \( \theta \) of a complex number \( a + bi \):

• Use the tangent function: \( \theta = \tan^{-1}(\frac{b}{a}) \)
• Be mindful of the quadrant: The signs of \( a \) and \( b \) determine which quadrant the angle is in.

This can be tricky, since angles can repeat after every 360° and calculators often provide only a principal value (usually between \(-90\degree\) and \(90\degree\)). For the exercise \(-\sqrt{21} + 10i\), the angle appears in the second quadrant, requiring an adjustment of the computed tangent angle by adding 180°, resulting in an argument of approximately 154°. This directional measure, combined with modulus, translates a rectangular form into polar form.