Problem 54 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$4\left(\cos 35^{\circ}+i \sin 35^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is approximately \( 3.2768 + 2.2944i \).

Step 1: Understanding the Problem

We need to express the complex number given in trigonometric form as a rectangular form (a + bi). The given form is: \( 4 (\cos 35^{\circ} + i \sin 35^{\circ}) \). We will retrieve the values of \( \cos 35^{\circ} \) and \( \sin 35^{\circ} \) and substitute them into the expression.

Step 2: Calculate Cosine and Sine

Use a calculator to find the actual values of \( \cos 35^{\circ} \) and \( \sin 35^{\circ} \). \( \cos 35^{\circ} \approx 0.8192 \)\( \sin 35^{\circ} \approx 0.5736 \)

Step 3: Substitute and Simplify

Substitute the values from Step 2 into the expression:\[ 4(\cos 35^{\circ} + i \sin 35^{\circ}) = 4(0.8192 + i \times 0.5736) \]Simplify this to get the rectangular form:\[ 4 \times 0.8192 + 4 \times 0.5736 i \]

Step 4: Final Calculation

Perform the multiplication:- For the real part: \( 4 \times 0.8192 = 3.2768 \)- For the imaginary part: \( 4 \times 0.5736 = 2.2944 \)Thus, the rectangular form is \( 3.2768 + 2.2944i \).

Key Concepts

Rectangular Form

The rectangular form of a complex number is a way to represent complex numbers as a sum of a real part and an imaginary part. It is written in the format \(a + bi\), where \(a\) is the real component and \(b\) is the imaginary component multiplied by \(i\), the imaginary unit. In this form:

• \(a\) represents the distance from the origin along the real axis.
• \(b\) (when multiplied by \(i\)) represents the distance along the imaginary axis.

This way of expressing complex numbers helps in direct arithmetic operations like addition and subtraction. In our solution, the complex number in rectangular form is \(3.2768 + 2.2944i\). This means:

• Real part: 3.2768
• Imaginary part: 2.2944

Together they form the complex number's position on the complex plane.

Trigonometric Form

The trigonometric form is an alternative method of expressing complex numbers using polar coordinates. Instead of specifying horizontal and vertical distances from the origin, a complex number is represented with a magnitude and an angle. This is especially useful in contexts involving rotation and scaling. It is typically expressed as:\[ r(\cos \theta + i \sin \theta) \]where:

• \(r\) is the magnitude or modulus of the complex number, calculated as \(\sqrt{a^2 + b^2}\).
• \(\theta\) is the angle or argument, computed from the arctangent of \(b/a\).

In our example, the original complex number is \(4(\cos 35^\circ + i \sin 35^\circ)\), which means:

• Magnitude \(r = 4\)
• Angle \(\theta = 35^\circ\)

Using this form is advantageous in multiplication and division of complex numbers, as it simplifies to multiplying magnitudes and adding angles.

Cosine and Sine Calculations

Cosine and sine functions are fundamental in converting trigonometric forms into rectangular forms. They relate to the horizontal and vertical components of a complex number on the complex plane. To convert a number in trigonometric form to rectangular form, you need:

• \(\cos \theta\) to adjust the real component.
• \(\sin \theta\) to adjust the imaginary component.

For practical applications like the exercise, it's common to use a calculator to determine these values:

• \(\cos 35^\circ \approx 0.8192\)
• \(\sin 35^\circ \approx 0.5736\)

These calculated values are then substituted back into the expression \(r(\cos \theta + i \sin \theta)\) to get it into the \(a + bi\) format. This is how we derived the rectangular form \(3.2768 + 2.2944i\) from the initial trigonometric form.