Problem 23 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=7 \sqrt{2} \operatorname{cis}\left(\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular form is \(z = 7 + 7i\).

Step 1: Understand the polar form

The given complex number is in the polar form, represented as \(z = r \operatorname{cis}(\theta)\), where \(r\) is the magnitude, and \(\theta\) is the argument of the complex number in radians. The formula for rectangular form is \(z = r(\cos \theta + i\sin \theta)\).

Step 2: Identify the values of \(r\) and \(\theta\)

From the given complex number \(z = 7 \sqrt{2} \operatorname{cis}(\frac{\pi}{4})\), we identify \(r = 7\sqrt{2}\) and \(\theta = \frac{\pi}{4}\).

Step 3: Calculate \(\cos(\theta)\) and \(\sin(\theta)\)

Using the known values, find \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). These are standard trigonometric values for the angle \(\frac{\pi}{4}\).

Step 4: Substitute and calculate the rectangular form

Substitute the values into the formula: \(z = 7\sqrt{2} (\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2})\). Simplify the expression: \(z = 7 \sqrt{2} \times \frac{\sqrt{2}}{2} + 7 \sqrt{2} \times i \frac{\sqrt{2}}{2}\). \(z = 7 + 7i\).

Step 5: Final Solution

The rectangular form of the given complex number is \(z = 7 + 7i\).

Key Concepts

Understanding Polar Form

Complex numbers can be expressed in different forms, with the polar form being one of the most common representations, apart from the rectangular form. In polar form, a complex number is represented as \( z = r \operatorname{cis}(\theta) \). Here, \( r \) represents the magnitude or modulus, which signifies the distance from the origin to the point in the complex plane. On the other hand, \( \theta \) is the argument, representing the angle made with the positive x-axis.

Polar form is particularly useful in scenarios involving geometric interpretations or calculations involving rotations and magnitudes, as it leverages trigonometry. It makes multiplying and dividing complex numbers more intuitive, as magnitudes and angles can be directly used to compute the results.

• **Magnitude **: \( r \), calculated as \( \sqrt{a^2 + b^2} \)
• **Argument**: \( \theta \), found through \( \tan^{-1} (\frac{b}{a}) \) for a complex number \( z = a + bi \)

By understanding polar form, converting to and from the rectangular form becomes much easier.

Using Trigonometric Identities

Trigonometric identities play a crucial role when dealing with the conversion between forms of complex numbers. In our context, the polar form of a complex number entails using the expression \( \operatorname{cis}(\theta) = \cos(\theta) + i\sin(\theta) \). This representation allows us to break down complex numbers into real and imaginary components directly.

**Key Trigonometric Identities:**

• **Cosine**: \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• **Sine**: \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)

These identities are essential for calculating the rectangular form. They allow us to express the complex number using real numbers and imaginary coefficients. Particularly, for converting from polar to rectangular, it's pivotal to compute the cosine and sine of the angle \( \theta \) given. Applying these trigonometric identities simplifies the transformation process making it seamless.

Complex Number Conversion

Convert complex numbers between different forms might seem challenging at first, but once you understand the mechanism, it becomes straightforward. The polar form to rectangular form conversion relies on the formula:
\[ z = r(\cos(\theta) + i\sin(\theta)) \]This formula emerges from the relation between our polar coordinates \(r\) and \(\theta\) to the classic x and y coordinates in the rectangular form \(z = x + yi\). The steps involve substituting the known values of \(r\), \(\theta\), \(\cos(\theta)\), and \(\sin(\theta)\) into this formula, effectively transforming the problem of operating on complex numbers into simple arithmetic.

For example, to convert \( z = 7\sqrt{2} \operatorname{cis}(\frac{\pi}{4}) \) to rectangular form, identify:

• **Magnitude**: \( r = 7\sqrt{2} \)
• **Angle**: \( \theta = \frac{\pi}{4} \)
• Substitute into formula for rectangular representation
• Use trigonometric identities to replace cos and sin values
• Simplify to get \( z = 7 + 7i \)

Thus, converting complex numbers is mostly about recognizing and calculating using the correct formulas and identities.