Problem 51 - Contemporary Precalculus

Question

In Exercises \(37-52,\) express the number in polar form. $$-\frac{5}{2}+\frac{7}{2} i$$

Step-by-Step Solution

Answer

Question: Convert the complex number \(-\frac{5}{2}+\frac{7}{2}i\) to its polar form. Answer: \(\frac{\sqrt{74}}{2}(\cos(\arctan(-\frac{7}{5})+\pi)+i\sin(\arctan(-\frac{7}{5})+\pi))\)

Step 1: Calculate the magnitude r

To calculate the magnitude r, use the formula \(\|z\|=\sqrt{(\Re(z))^2+(\Im(z))^2}\): $$r=\sqrt{(-\frac{5}{2})^2+(\frac{7}{2})^2}=\sqrt{\frac{74}{4}} =\frac{\sqrt{74}}{2}$$

Step 2: Determine the angle θ

To find θ, use the formula \(\tan(\theta)=\frac{\Im(z)}{\Re(z)}\). First, find the arctangent of the ratio: $$\tan(\theta)=\frac{7/2}{-5/2}=-\frac{7}{5}$$ This gives us: $$\theta=\arctan(-\frac{7}{5})$$ As the complex number is in the second quadrant, add \(\pi\) to the angle: $$\theta=\arctan(-\frac{7}{5}) + \pi$$

Step 3: Express the polar form

Now, substitute r and θ to the polar form equation \(r(\cos(\theta)+i\sin(\theta))\): $$-\frac{5}{2}+\frac{7}{2}i=\frac{\sqrt{74}}{2}(\cos(\arctan(-\frac{7}{5})+\pi)+i\sin(\arctan(-\frac{7}{5})+\pi))$$

Key Concepts

Complex numbers

Complex numbers are an extension of the number system we use every day, known as real numbers. They include a real part and an imaginary part. The imaginary part is defined as a real number multiplied by the imaginary unit 'i', which is the square root of \( -1 \). In this exercise, the number \(-\frac{5}{2}+\frac{7}{2}i\) is a complex number where:

• The real part is \(-\frac{5}{2}\)
• The imaginary part is \(\frac{7}{2}\)

Complex numbers are often visualized in the complex plane with the real part represented on the x-axis and the imaginary part on the y-axis. This allows us to develop further concepts such as their magnitude and angle.

Magnitude

The magnitude of a complex number is akin to measuring the length of its vector from the origin in the complex plane. It's calculated using the formula for the Euclidean distance: \[ \|z\| = \sqrt{(\Re(z))^2 + (\Im(z))^2} \]For our given complex number \(-\frac{5}{2}+\frac{7}{2}i\), the magnitude is computed as follows:

• The square of the real part: \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\)
• The square of the imaginary part: \(\left(\frac{7}{2}\right)^2 = \frac{49}{4}\)

Adding these gives us:\[ \|z\| = \sqrt{ \frac{25}{4} + \frac{49}{4} } = \sqrt{ \frac{74}{4} } = \frac{\sqrt{74}}{2} \]The magnitude tells us directly about the size of the complex number, without regard to its angle or direction.

Angle

The angle of a complex number in polar coordinates, also known as the argument, determines in which direction the vector representing the complex number points from the origin. It is found using the arctangent and the coordinates of the point on the complex plane:For \(-\frac{5}{2} + \frac{7}{2}i\), this means calculating:\[ \tan(\theta) = \frac{\Im(z)}{\Re(z)} = \frac{7/2}{-5/2} = -\frac{7}{5} \]This results in:\[ \theta = \arctan\left(-\frac{7}{5}\right) \]Since our complex number is situated in the second quadrant (as the real part is negative and imaginary part positive), we adjust the angle by adding \(\pi\) radians:\[ \theta = \arctan\left(-\frac{7}{5}\right) + \pi \]This adjustment ensures our angle accurately reflects the complex number's position.

Arctangent

The arctangent function, often denoted as \(\arctan\), is used to find an angle whose tangent is a given value. It is the inverse function of the tangent. Applicable in many scenarios, it helps ascertain the angle a vector has in the complex coordinate plane.For the complex number \(-\frac{5}{2}+\frac{7}{2}i\), we apply it to the ratio of the imaginary to the real components:\[ \arctan\left(-\frac{7}{5}\right) \]This function yields a principal value between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). However, because the complex number is in the second quadrant, we add \(\pi\) to derive the true angle without representation errors in different quadrants, ensuring accurate direction on the complex plane.