Problem 36 - Algebra and Trigonometry

Question

In Exercises \(27-36,\) write each complex number in rectangular form. If necessary, round to the nearest tenth. $$ 30(\cos 2.3+i \sin 2.3) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number \(30(\cos 2.3+i \sin 2.3)\) is \(-24 + 18i\).

Step 1: Identify the complex number in polar form

The complex number in polar form is given as \(30(\cos 2.3+i \sin 2.3)\), where \(r=30\) represents the modulus and \(2.3\) represents the argument in radians.

Step 2: Apply the identities for cosine and sine

By applying the standard cosine and sine function on the calculator, you will get that \(\cos 2.3 \approx -0.8\) and \(\sin 2.3 \approx 0.6 \) rounded to the nearest whole number.

Step 3: Calculate rectangular form

The rectangular form is obtained by multiplying the modulus with both the cosine and sine of the complex number, denoted by a and b respectively, which are \(a=r\cos \theta\) and \(b=r\sin \theta\). Here, \(a=30 * -0.8 = -24\) and \(b=30 * 0.6 = 18\).

Step 4: Substitute the values

The rectangular form of the complex number is \(a + bi\). After calculating the values of a and b, substitute these values to obtain: \(-24 + 18i\).

Key Concepts

Rectangular Form

Complex numbers can be expressed in several ways, but one common method is the rectangular form. This form represents a complex number as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In the rectangular form, each complex number is plotted on a two-dimensional plane, making it easy to visualize.

• The real component \(a\) is the projection on the horizontal axis.
• The imaginary component \(b\), accompanied by the imaginary unit \(i\), is the projection on the vertical axis.

Understanding the rectangular form is crucial for performing operations like addition, subtraction, and multiplication of complex numbers. It provides a straightforward way to handle these operations, using familiar arithmetic rules.

Polar Form

The polar form is another way to express complex numbers. Instead of focusing on the real and imaginary components, this form highlights the distance from the origin and the angle with the positive real axis. In polar form, a complex number is given as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

• The modulus \( r \) indicates how far the point is from the origin.
• The argument \( \theta \) shows the direction of the point from the positive real axis.

Polar form is particularly useful when multiplying or dividing complex numbers, allowing these operations to be carried out more simply by manipulating angles and moduli. Converting from polar to rectangular form involves calculating \( r \cos \theta \) and \( r \sin \theta \) to get the real and imaginary parts.

Modulus and Argument

Understanding modulus and argument is key to working with complex numbers in polar form. The modulus, \(r\), represents the "length" of the vector from the origin to the point. It can be calculated as \(\sqrt{a^2 + b^2}\) when you're given a rectangular form \(a + bi\). The argument, \(\theta\), is the angle made with the positive real axis. It is typically measured in radians.

• To find \(\theta\), use the arctangent function: \(\theta = \arg(\frac{b}{a})\).
• The argument gives the direction of the vector relative to the positive x-axis.

These two components provide a comprehensive description of the position of a complex number on the complex plane, facilitating conversions between polar and rectangular forms.

Sine and Cosine Identities

Sine and cosine identities are pivotal when working with complex numbers, especially in polar form. These trigonometric functions help translate angle measures into real and imaginary components. For any angle \(\theta\):

• \(\cos \theta\) corresponds to the x-coordinate (real part).
• \(\sin \theta\) represents the y-coordinate (imaginary part).

In the context of complex numbers, using sine and cosine identities allows us to switch between rectangular and polar forms seamlessly. Moreover, these identities underpin the solutions of problems involving periodicity and oscillations, making them not just a tool for conversions but a foundational principle of trigonometry. These functions make the calculation of the real and imaginary parts from a given polar form straightforward by multiplying the modulus by \(\cos \theta\) and \(\sin \theta\), respectively.