Problem 31 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\frac{1}{2} \operatorname{cis}\left(\frac{7 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \( \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4}i \).

Step 1: Understand the given complex number

The given complex number is in polar form, represented as \ \( z = r \text{cis}(\theta) = r(\cos(\theta) + i\sin(\theta)) \). Here, \ \( r = \frac{1}{2} \) and \ \( \theta = \frac{7\pi}{4} \). Our goal is to convert this into rectangular form, which is \ \( a + bi \).

Step 2: Calculate the cosine component

Substitute \ \( \theta = \frac{7\pi}{4} \) into the cosine function: \ \( \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Thus, the real part is \ \( \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \).

Step 3: Calculate the sine component

Substitute \ \( \theta = \frac{7\pi}{4} \) into the sine function: \ \( \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). Thus, the imaginary part is \ \( \frac{1}{2} \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{4} \).

Step 4: Formulate the rectangular form

Combine the real and imaginary parts to express the complex number in rectangular form: \ \( z = \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4}i \).

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be represented in various forms, one of which is the polar form. In polar form, a complex number is written as \( z = r \operatorname{cis}(\theta) = r(\cos(\theta) + i\sin(\theta)) \). Here, \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the angle (or argument) measured in radians.
To visualize, imagine plotting the complex number on a plane. The distance from the origin to the point is \( r \), while \( \theta \) is the angle formed with the positive x-axis. This form is especially useful when performing operations like multiplication or division, which are more cumbersome in rectangular form.
By expressing complex numbers as products of their magnitude and angle, calculations involving trigonometry become more manageable.

Cosine and Sine Components

In polar form, the expression \( r(\cos(\theta) + i\sin(\theta)) \) breaks down into two parts: the cosine component \( r\cos(\theta) \) and the sine component \( r\sin(\theta) \). These components give the real and imaginary parts, respectively, for the conversion to rectangular form.

• The cosine term provides the real part: \( \cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2} \). Multiplying this with \( r = \frac{1}{2} \) results in the real part as \( \frac{\sqrt{2}}{4} \).
• The sine term provides the imaginary part: \( \sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2} \). Similarly, multiplying this value with \( r \) offers the imaginary part as \( -\frac{\sqrt{2}}{4}i \).

Both components are crucial for translating between polar and rectangular forms accurately.

Conversion to Rectangular Form

Converting a complex number from its polar form to its rectangular form involves finding its real and imaginary components. This allows a transformation from \( z = r(\cos(\theta) + i\sin(\theta)) \) to the standard rectangular form \( a + bi \), where \( a \) and \( b \) are real numbers.
The process is straightforward once you understand the cosine and sine components:

• Calculate the real part by multiplying \( r \) with \( \cos(\theta) \).
• Evaluate the imaginary part by multiplying \( r \) with \( \sin(\theta) \).
• Combine both to get the rectangular form: \( a + bi = \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4}i \).

This method provides a straightforward route to understanding and applying the two sides of complex number representation.

Exact Values in Trigonometry

In trigonometry, knowing exact values for certain angles is key in problem-solving. With angles like \( \frac{7\pi}{4} \), exact values can be obtained using reference angles and the unit circle.
The unit circle helps you to find sine and cosine values without a calculator by noting that:

• \( \cos(\frac{7\pi}{4}) \) corresponds to \( \cos(45^\circ) \), which is \( \frac{\sqrt{2}}{2} \).
• \( \sin(\frac{7\pi}{4}) \) points downward from the x-axis, yielding \( -\frac{\sqrt{2}}{2} \).
• These values are derived due to the symmetrical nature of the trigonometric functions on the unit circle.

Understanding these exact values is essential as it simplifies the process of conversion between polar and rectangular forms, ensuring accuracy in calculations.