Problem 72 - Precalculus

Question

Explain the mistake that is made. Express \(z=-3+8 i\) in polar form. Solution: $$\begin{array}{ll}\text { Find } r . & r=\sqrt{x^{2}+y^{2}}=\sqrt{9+64}=\sqrt{73} \\\\\text { Find } \theta . & \tan \theta=-\frac{8}{3}\end{array}$$ $$\theta=\tan ^{-1}\left(-\frac{8}{3}\right)=-69.44^{\circ}$$ Write the complex number in polar form. $$z=\sqrt{73}\left[\cos \left(-69.44^{\circ}\right)+i \sin\left(-69.44^{\circ}\right)\right]$$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

Mistake in quadrant identification; corrected angle is \(110.56^{\circ}\).

Step 1: Identify the Given Complex Number

The given complex number is \( z = -3 + 8i \), where \( x = -3 \) and \( y = 8 \).

Step 2: Calculate the Magnitude (r)

Calculate the magnitude using the formula \( r = \sqrt{x^2 + y^2} \). Substituting \( x = -3 \) and \( y = 8 \), we have: \[ r = \sqrt{(-3)^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73}. \] The calculation of \( r \) is correct.

Step 3: Calculate the Argument (θ) Incorrectly

The argument \( \theta \) is calculated incorrectly. The formula used is \( \tan \theta = \frac{y}{x} \). Substituting \( y = 8 \) and \( x = -3 \), the calculation should be: \[ \tan \theta = \frac{8}{-3} = -\frac{8}{3}. \] This calculation gives an angle \( \theta = \tan^{-1}\left(-\frac{8}{3}\right), \) but the resulting angle is \(-69.44^{\circ}\), which suggests a mistake in quadrant identification.

Step 4: Correct the Angle for the Appropriate Quadrant

Given that the complex number \( z = -3 + 8i \) is in the second quadrant (since \( x < 0 \) and \( y > 0 \)), the correct angle \( \theta \) should be adjusted. The angle \( \theta \) should be:\[ \theta = 180^{\circ} + \tan^{-1}\left(-\frac{8}{3}\right) = 180^{\circ} - 69.44^{\circ} = 110.56^{\circ}. \]

Step 5: Express the Complex Number in Polar Form

The correct polar form of the complex number is:\[ z = \sqrt{73}\left[ \cos(110.56^{\circ}) + i \sin(110.56^{\circ}) \right]. \]

Key Concepts

Magnitude of a Complex Number

The magnitude of a complex number is a measure of its size or distance from the origin on the complex plane. This is the same concept as the hypotenuse of a right triangle. To find the magnitude of any complex number given in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, you use the formula, \( r = \sqrt{a^2 + b^2} \). This formula is derived from Pythagoras' theorem.

For example, if you have \( z = -3 + 8i \), the magnitude \( r \) is calculated by taking the square root of the sum of the squares of its real part \( -3 \) and its imaginary part \( 8 \). This gives:

• \( r = \sqrt{(-3)^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73}. \)

Understanding this concept is crucial because the magnitude tells us how far away from the origin the complex number is, much like the distance between two points in the field of geometry.

Argument of a Complex Number

Finding the argument of a complex number involves calculating the angle it makes with the positive real axis, measured in a counterclockwise direction. This angle is often denoted by \( \theta \). The formula used to find \( \theta \) when given \( z = a + bi \) is \( \tan \theta = \frac{b}{a} \). It provides the tangent of the angle, which can then be used to find the angle itself using the inverse tangent function, \( \tan^{-1} \).

However, while using this formula, it's important to consider the signs of \( a \) and \( b \) to determine the correct quadrant for \( \theta \). For the complex number \( z = -3 + 8i \), \( y = 8 \) and \( x = -3 \), giving \( \tan \theta = \frac{8}{-3} = -\frac{8}{3} \). An initial calculation might give \( \theta = \tan^{-1}\left( -\frac{8}{3} \right) = -69.44^{\circ} \).

• However, since \( a < 0 \) and \( b > 0 \), the complex number actually lies in the second quadrant.
• This necessitates an adjustment to the angle, resulting in \( \theta = 180^{\circ} - 69.44^{\circ} = 110.56^{\circ} \).

The argument helps us in deciding where precisely the complex number is located on the complex plane, by establishing its direction relative to the positive real axis.

Quadrants in Complex Numbers

Just like in the Cartesian coordinate system, the complex plane is divided into four quadrants. These quadrants help in determining the direction and position of a complex number based on its real and imaginary parts.

Each quadrant consists of combinations of signs (positive or negative) of the real part \( x \) and the imaginary part \( y \):

• The first quadrant has both real and imaginary parts positive \((x > 0, y > 0)\).
• The second quadrant has a negative real part and a positive imaginary part \((x < 0, y > 0)\).
• The third quadrant contains both parts negative \((x < 0, y < 0)\).
• The fourth quadrant has a positive real part and a negative imaginary part \((x > 0, y < 0)\).

Understanding quadrants is essential for determining the correct argument of a complex number, as seen in the example of \( z = -3 + 8i \). Its location in the second quadrant means the argument \( \theta \) needs adjusting. This helps accurately represent the complex number in its correct position with respect to the axes and reflect the sign changes in the trigonometric calculations.