Problem 53 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$5\left(\cos 295^{\circ}+i \sin 295^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is approximately \( 2.113 - 4.5315i \).

Step 1: Understand the Polar Form

The given complex number is in polar form. A complex number in polar form is written as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus, and \( \theta \) is the angle. In this problem, \( r = 5 \) and \( \theta = 295^{\circ} \).

Step 2: Determine the Rectangular Coordinates

To convert from polar to rectangular form, compute the real part as \( r \cos \theta \) and the imaginary part as \( r \sin \theta \). We need to find these values using the given \( \theta = 295^{\circ} \).

Step 3: Calculate \( \cos 295^{\circ} \)

Using a calculator, determine \( \cos 295^{\circ} \). Recall that \( 295^{\circ} \) is in the fourth quadrant, where cosine is positive. \( \cos 295^{\circ} \approx 0.4226 \).

Step 4: Calculate \( \sin 295^{\circ} \)

Using a calculator, determine \( \sin 295^{\circ} \). Recall that \( 295^{\circ} \) is in the fourth quadrant, where sine is negative. \( \sin 295^{\circ} \approx -0.9063 \).

Step 5: Multiply with Modulus

Compute the rectangular form by multiplying each trigonometric result by the modulus \( r = 5 \). The real part is \( 5 \times 0.4226 = 2.113 \). The imaginary part is \( 5 \times (-0.9063) = -4.5315 \).

Step 6: Express in Rectangular Form

Combine the real and imaginary parts to express the complex number in rectangular form: \( 2.113 - 4.5315i \).

Key Concepts

Polar Form

In mathematics, complex numbers can be expressed using polar form, a method which utilizes the modulus and angle to represent the number. This form is particularly useful because it simplifies many operations, such as multiplication and division.

• The polar form is written as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus.
• \( \theta \) is the argument, an angle in degrees or radians that represents the direction.
• This powerful form highlights the geometric interpretation of complex numbers, aligning them with vectors in the complex plane.

Understanding the angle \( \theta \) and how it influences the position in the plane helps in visualizing complex numbers during operations, making them easier to manipulate and understand.

Rectangular Form

Rectangular form, also known as the Cartesian form, expresses complex numbers using real and imaginary components. This is the typical representation as it mirrors how points are plotted on a 2-dimensional graph.

• The general expression is \( a + bi \), where \( a \) represents the real part.
• \( b \) is the coefficient of the imaginary unit \( i \), addressing the imaginary part.

This form makes addition and subtraction straightforward as you can directly combine the real parts while separately combining the imaginary parts. Converting from polar to rectangular form involves finding the equivalent coordinates (real and imaginary) by calculating \( a = r \cos \theta \) and \( b = r \sin \theta \). This transition helps in diverse applications, including engineering and physics, where direct manipulation of real numbers is necessary.

Modulus

The modulus of a complex number is an essential feature that defines its size or absolute value. It is pivotal in the conversion between polar and rectangular forms.

• For a complex number \( z = a + bi \), the modulus is defined as \( |z| = \sqrt{a^2 + b^2} \).
• In polar form, \( r \) represents the modulus, which is the distance from the origin to the point \((a, b)\) in the complex plane.

Having grasped the modulus helps in many operations, such as finding the root of a complex number or expressing it in a trigonometric form. It essentially gauges how far the number is from the origin, likening it to the length of a vector.

Trigonometric Functions

Trigonometric functions play a pivotal role in the process of converting complex numbers from polar to rectangular form. They help find the specific coordinates on the graph.

• These functions are sine (\( \sin \)) and cosine (\( \cos \)), which are fundamental in either estimating or calculating the real and imaginary parts.
• For example, the \( \cos \theta \) helps determine the real part, and \( \sin \theta \) aids in calculating the imaginary part.

Understanding the trigonometric ratios, especially in different quadrants, is crucial as these affect the values. For values like \( 295^\circ \), which lie in the fourth quadrant, the cosine remains positive while the sine is negative. This insight is imperative when expressing the complex number accurately from polar to rectangular form.