Problem 28 - Algebra and Trigonometry

Question

In Exercises \(27-36,\) write each complex number in rectangular form. If necessary, round to the nearest tenth. $$ 12\left(\cos 60^{\circ}+i \sin 60^{\circ}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number \(12(\cos 60^{\circ}+i \sin 60^{\circ})\) is \(6 + 10.4i\).

Step 1: Calculate x

Calculate the x-coordinate (real part) using the formula \(x = r\cos \theta\). Here, \(r = 12\) and \(\theta = 60^{\circ}\). So, \(x = 12\cos 60^{\circ}\). Cosine of \(60^{\circ}\) is \(0.5\), thus \(x = 12 * 0.5 = 6\).

Step 2: Calculate y

Calculate the y-coordinate (imaginary part) using the formula \(y = r\sin \theta\). Here, \(r = 12\) and \(\theta = 60^{\circ}\). So, \(y = 12\sin 60^{\circ}\). Sine of \(60^{\circ}\) is \(\sqrt{3}/2 \approx 0.866\), thus \(y = 12 * 0.866 \approx 10.4\).

Step 3: Write in Rectangular Form

The rectangular form of a complex number is \(x + iy\). After substituting \(x = 6\) and \(y = 10.4\) into this, we get the complex number in rectangular form as \(6 + 10.4i\).

Key Concepts

Rectangular Form

Rectangular form is a way to represent complex numbers using a combination of real numbers and imaginary numbers. This form is typically expressed as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary unit is denoted by \(i\), which is equal to the square root of -1.

• In rectangular form, complex numbers are easy to plot on the complex plane.
• The x-axis represents the real part, while the y-axis represents the imaginary part.

To convert from trigonometric to rectangular form, you use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\) to find the real and imaginary components.

Trigonometric Form

The trigonometric form of a complex number is an alternative representation that uses polar coordinates. It expresses a complex number in the form \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the angle (or argument).

• Magnitude \(r\) represents the distance from the origin to the point on the complex plane.
• Angle \(\theta\) is typically measured in degrees or radians from the positive real axis.

This form is especially useful for multiplying or dividing complex numbers, as it simplifies operations through angle addition or subtraction.

Real Part

The real part of a complex number is the component that does not involve the imaginary unit \(i\). For a complex number \(a + bi\), the real part is \(a\). It is similar to a regular, non-complex number that you would find on the real number line.

• The real part can be visualized as the location along the x-axis on a complex plane.
• In problems involving trigonometric form, it is calculated using the formula: \(x = r \cos \theta\).

Understanding the real part helps in decomposing a complex number into its constituent components, making addition and subtraction straightforward.

Imaginary Part

The imaginary part of a complex number is what accompanies the imaginary unit \(i\), which is used to extend the real number line into a complex plane. For a complex number \(a + bi\), the imaginary part is \(b\). This represents the vertical component when represented graphically.

• The imaginary part can be thought of as the displacement along the y-axis on a complex plane.
• Calculated in trigonometric form using \(y = r \sin \theta\), it allows one to visualize oscillations and rotations that occur at the imaginary dimension.

The imaginary part is crucial for representing wave patterns and in solving equations that otherwise do not intersect the real number line.