Problem 61 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$-2\left[\cos \left(\frac{3 \pi}{5}\right)+i \sin \left(\frac{3 \pi}{5}\right)\right]$$

Step-by-Step Solution

Answer

The complex number in rectangular form is approximately \(0.618 - 1.902i\).

Step 1: Identify the Given Complex Number

The complex number is given in polar form: \(-2\left[\cos \left(\frac{3 \pi}{5}\right)+i \sin \left(\frac{3 \pi}{5}\right)\right]\). This can be expressed as \(r\left[\cos \theta + i \sin \theta\right]\) with \(r = -2\) and \(\theta = \frac{3 \pi}{5}\).

Step 2: Calculate the Rectangular Form

To convert from polar to rectangular form, use the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). First, calculate \(x = -2 \cos \left(\frac{3 \pi}{5}\right)\): - Use a calculator to find \(\cos \left(\frac{3 \pi}{5}\right)\) which is approximately \(-0.309\).- Therefore, \(x = -2 \, \times\, -0.309 = 0.618\).Next, calculate \(y = -2 \sin \left(\frac{3 \pi}{5}\right)\):- Use a calculator to find \(\sin \left(\frac{3 \pi}{5}\right)\) which is approximately \(0.951\).- Therefore, \(y = -2 \, \times\, 0.951 = -1.902\).

Step 3: Write the Complex Number in Rectangular Form

Combine the calculated values of \(x\) and \(y\) to express the complex number in rectangular form: \(0.618 - 1.902i\).

Key Concepts

Polar to Rectangular Conversion

When dealing with complex numbers, they can be expressed in two main forms: polar and rectangular forms. Polar form is particularly useful when dealing with trigonometry and complex number multiplication or division. To convert a complex number from polar to rectangular form, you need to use a few simple formulas. These formulas involve breaking down the complex number into its real and imaginary components using its magnitude and angle. The polar form of a complex number is given as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the angle (or argument).

• To convert to rectangular form, the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \) are used. Here, \( x \) represents the real part, and \( y \) represents the imaginary part of the complex number.
• This conversion is often used in contexts where addition or subtraction of complex numbers is required, since rectangular form simplifies these operations.

By applying these formulas, you can easily switch between the two representations, allowing for greater flexibility in using complex numbers in various mathematical and engineering applications.
Converting complex numbers from polar to rectangular form simplifies the process of calculating and visualizing complex equations and performs practical calculations in areas such as electrical engineering.

Trigonometric Form of Complex Numbers

The trigonometric form offers a way to express complex numbers by leveraging their geometric interpretation. A complex number can be visually represented in the complex plane, with the real part on the x-axis and the imaginary part on the y-axis. This makes the trigonometric form particularly intuitive for understanding the position and rotation of complex numbers in relation to the origin.

• The general form \( r(\cos \theta + i \sin \theta) \) allows you to understand a complex number as a point that is \( r \) units away from the origin, making an angle \( \theta \) with the positive x-axis.
• Unlike the standard rectangular \( a + bi \) form, the trigonometric form emphasizes the angle and distance from the origin, which can simplify problems involving powers and roots of complex numbers.

The trigonometric form becomes especially beneficial when dealing with multiplication and division of complex numbers. When multiplying two complex numbers in trigonometric form, their magnitudes are multiplied, and their angles are added, simplifying what could otherwise be a complex algebraic process.
This form is a staple in many educational curricula and practical applications, bridging the concepts of trigonometry and complex number theory.

Rectangular Form of Complex Numbers

The rectangular form is arguably the most straightforward way to express complex numbers, making it an essential concept for anyone learning about them. This form presents complex numbers in the canonical form \( a + bi \), where \( a \) and \( b \) are real numbers.

• Here, \( a \) is referred to as the real component, and \( bi \) is the imaginary component. The entire expression is understood as one number in the complex plane.
• This form is intuitive for operations such as addition and subtraction, as it aligns with regular arithmetic of real numbers.

Furthermore, rectangular form is accessible and useful for graphing complex numbers. By considering \( a \) as the x-coordinate and \( b \) as the y-coordinate, complex numbers can be easily plotted on the complex plane.
Many practical applications rely on the rectangular form to simplify processes, including in fields like electrical engineering and physics, where complex numbers frequently come into play. The juxtaposition of the real and imaginary parts in rectangular form allows for straightforward analysis and computation of complex situations.