Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\frac{\cos \left(\frac{7 \pi}{4}\right)+i \sin \left(\frac{7 \pi}{4}\right)}{\cos \pi+i \sin \pi}$$

Step-by-Step Solution

Answer

The result of the division operation in trigonometric form is \(\cos \left(-\frac{ \pi}{4}\right)+i \sin \left(-\frac{ \pi}{4}\right)\)

Step 1: Rewrite in Euler's Form

Recognize that the given expression is in Euler's form and rewrite it using exponential notation. So it becomes \(\frac{e^{i\frac{7 \pi}{4}}}{e^{i\pi}}\).

Step 2: Simplify

Recognize that division in exponential notation becomes subtraction in the exponent. So transform the above expression to \(e^{i(\frac{7 \pi}{4}-\pi)}\).

Step 3: Calculate Exponent

Subtract the exponents to get \(e^{i(-\frac{\pi}{4})}\). This is the answer in exponential form.

Step 4: Convert Back to Trigonometric Form

Using Euler's formula, we convert the result back to the trigonometric form, which is \(\cos \left(-\frac{ \pi}{4}\right)+i \sin \left(-\frac{ \pi}{4}\right)\).

Key Concepts

Euler's Formula

Euler's formula is a crucial concept in complex numbers and trigonometry. It links complex exponential functions with trigonometric functions. The formula is expressed as \(e^{i\theta} = \cos(\theta) + i \sin(\theta)\). Here, \(\theta\) represents the angle in radians, \(\cos(\theta)\) is the cosine of that angle, and \(i \sin(\theta)\) contains the imaginary unit \(i\).

This formula allows us to express complex numbers in exponential form, making complex arithmetic much simpler. For instance, in the exercise given, both the numerator and the denominator can be written using Euler's formula. The expression \(\cos \left(\frac{7\pi}{4}\right) + i \sin \left(\frac{7\pi}{4}\right)\) transforms into \(e^{i\frac{7\pi}{4}}\), and \(\cos \pi + i \sin \pi\) becomes \(e^{i\pi}\). Utilizing Euler's formula simplifies the division of these complex numbers.

Exponent Subtraction

Exponent subtraction is a straightforward and useful method when dealing with division in exponential form, especially with complex numbers. When dividing two exponential expressions with the same base, simply subtract the exponents. This concept is reflected in the formula:

• \( \frac{a^m}{a^n} = a^{m-n} \)

For our given problem, after rewriting using Euler's formula, the expression becomes \( \frac{e^{i\frac{7\pi}{4}}}{e^{i\pi}} \). Applying exponent subtraction, this simplifies to:

• \( e^{i(\frac{7\pi}{4} - \pi)} \)

Once the exponents are subtracted, the expression becomes \( e^{-i\frac{\pi}{4}} \). Simplifying the division of exponentials through subtraction reduces complexity and provides an elegant solution in the exponential form.

Cosine and Sine Expressions

Converting back to cosine and sine expressions from the exponential form involves using Euler's formula again. After performing exponent subtraction, we have \( e^{-i\frac{\pi}{4}} \). To bring this expression back to trigonometric form, apply:

• \(e^{i\theta} = \cos(\theta) + i \sin(\theta)\)

The resulting expression can be converted into \( \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right) \).

The angle \(-\frac{\pi}{4}\) corresponds to a rotation in the counterclockwise direction, leading to values that are familiar. Cosine and sine of \(-\frac{\pi}{4}\) derive from their unit circle values:

• \( \cos\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
• \( \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \)

Transforming back to trigonometric form allows you to easily interpret the equivalent angle using the unit circle.