Problem 35 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=5 \operatorname{cis}\left(\arctan \left(\frac{4}{3}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \(z = 3 + 4i\).

Step 1: Understanding 'cis' Notation

The notation \(\operatorname{cis}(\theta)\) stands for \(\cos(\theta) + i\sin(\theta)\). The complex number \(z = 5 \operatorname{cis}\left(\arctan\left(\frac{4}{3}\right)\right)\) can be rewritten in rectangular form as \(z = 5(\cos(\theta) + i\sin(\theta))\), where \(\theta = \arctan\left(\frac{4}{3}\right)\).

Step 2: Calculate the Angle \(\theta\)

Since \(\theta = \arctan\left(\frac{4}{3}\right)\), \(\theta\) is the angle of a right triangle with opposite side 4 and adjacent side 3. Using the Pythagorean theorem, the hypotenuse \(r\) is \(\sqrt{4^2 + 3^2} = 5\).

Step 3: Find \(\cos(\theta)\)

For the angle \(\theta\), \(\cos(\theta)\) is the ratio of the adjacent side to the hypotenuse. Thus, \(\cos(\theta) = \frac{3}{5}\).

Step 4: Find \(\sin(\theta)\)

Similarly, \(\sin(\theta)\) is the ratio of the opposite side to the hypotenuse. Thus, \(\sin(\theta) = \frac{4}{5}\).

Step 5: Substitute into Rectangular Form

Now, substitute \(\cos(\theta)\) and \(\sin(\theta)\) into the rectangular form: \(z = 5(\frac{3}{5} + i\frac{4}{5})\).

Step 6: Simplify the Expression

Simplify the expression by distributing the 5: \(z = 5 \times \frac{3}{5} + 5 \times i \frac{4}{5} = 3 + 4i\).

Key Concepts

Rectangular Form

Rectangular form is a way of expressing complex numbers using real and imaginary components. This form is essential for performing arithmetic operations with complex numbers. In rectangular form, a complex number takes the structure

• \(a + bi\)

where \(a\) represents the real part, and \(bi\) represents the imaginary part.
This method of representation is straightforward and visually clear, making it easy to plot complex numbers on a two-dimensional graph. On this graph, the x-axis corresponds to the real part and the y-axis corresponds to the imaginary part of the complex number.
Together, these axes form a plane known as the complex plane, which helps in visualizing these numbers effectively.

Complex Numbers

Complex numbers are an extension of real numbers and provide a framework for solving equations that do not have real solutions. A complex number is composed of a real part and an imaginary part. The imaginary part includes the square root of negative one, represented as \(i\). Complex numbers are particularly useful in various fields such as engineering, physics, and mathematics. They can be represented as

• \(a + bi\)

where \(a\) and \(b\) are real numbers.
These numbers facilitate the expression of equations that have no real solutions. For instance, the solution to the equation \(x^2 + 1 = 0\) is \(\pm i\), as it incorporates the imaginary unit \(i\). In essence, complex numbers extend the real number system, allowing for the comprehensive solving of equations that would otherwise be insoluble.

Cis Notation

Cis notation offers a compact way to represent complex numbers using their magnitude and direction. The notation \(\operatorname{cis}(\theta)\) stands for

• \(\cos(\theta) + i\sin(\theta)\)

This form is particularly useful in polar representation where complex numbers are expressed in terms of a radius \(r\) and angle \(\theta\).
For example, a complex number of the form \(r \operatorname{cis}(\theta)\) can be translated into rectangular form by multiplying the radius by the cosine and sine components:

• \(r(\cos(\theta) + i\sin(\theta))\)

This approach makes it easier to handle multiplicative operations and powers of complex numbers, as it leverages the exponential form of complex numbers, which is

• \(re^{i\theta}\)

Trigonometric Functions

Trigonometric functions such as sine and cosine are vital in working with complex numbers, especially when dealing with angles and transformations. These functions are derived from the unit circle and are key in converting between rectangular and polar forms.
The cosine of an angle \(\theta\) is the x-coordinate, while the sine of the angle is the y-coordinate on the unit circle. These coordinates form the basis of translating polar coordinates into rectangular form, as seen in

• \(z = r(\cos(\theta) + i\sin(\theta))\)

Calculating trigonometric functions involves determining the ratios of sides in a right triangle, allowing for the expression of angles and lengths. These functions form the foundation for solving many mathematical problems, including the conversion between complex number forms and solving real-world problems involving oscillations and waves.