Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\cos \theta|-\ln |\sin \theta|$$

Step-by-Step Solution

Answer

The simplified single logarithm of the original expression is \( \ln|\cot \theta| \)

Step 1: Apply the property of logarithms

The expression should be rewritten using the properties of logarithms. Since we have subtraction taking place between two logarithms, this can be transformed into division. Thus, \( \ln|\cos \theta| - \ln|\sin \theta| \) becomes \( \ln\left|\frac{\cos \theta}{\sin \theta}\right| \).

Step 2: Simplify the expression

The division expression \( \frac{\cos \theta}{\sin \theta} \) can be simplified to \( \cot \theta \), so the final answer is \( \ln|\cot \theta| \)

Key Concepts

Logarithmic Identities

Logarithmic identities provide a set of rules that are extremely helpful in simplifying expressions involving logarithms. They allow us to manipulate expressions so they are easier to work with. One of the most commonly used properties is the difference of logarithms, which states:

• Logarithmic Difference: \( \log_b{A} - \log_b{B} = \log_b{\frac{A}{B}} \)

Here, \( A \) and \( B \) must be positive numbers due to the domain of logarithmic functions.
This identity transforms the subtraction of two logarithms into the logarithm of a division, simplifying complex logarithmic expressions. It plays a crucial role when dealing with ratios and simplifying calculations. In our exercise, this identity is used to transform the expression \( \ln|\cos \theta| - \ln|\sin \theta| \) into the simpler form of a single logarithm \( \ln\left|\frac{\cos \theta}{\sin \theta}\right| \).

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, particularly when dealing with angles and their relationships. They correspond to specific ratios of sides in right-angle triangles. The most basic trigonometric functions are sine (\( \sin \)) and cosine (\( \cos \)).

• Sine of an angle \( \theta \) is represented by \( \sin \theta \)
• Cosine of an angle \( \theta \) is represented by \( \cos \theta \)

These functions are periodic and can be applied to measure the cyclic properties of angles.
In this context, these trigonometric ratios \( \cos \theta \) and \( \sin \theta \) are crucial for expressing the original function before simplifying it. In our problem, \( \frac{\cos \theta}{\sin \theta} \) is formatted as a single function \( \cot \theta \), which is the cotangent function, a less common but equally important trigonometric identity used to express angle relationships.

Simplification of Logarithms

Simplifying logarithms includes combining them into a single expression and possibly transforming them into simpler functions. This involves using logarithmic properties to make calculations more manageable.
To simplify our expression \( \ln|\cos \theta| - \ln|\sin \theta| \), the difference property is first utilized, as previously mentioned. Then, we further simplify the resulting expression \( \ln\left|\frac{\cos \theta}{\sin \theta}\right| \) by recognizing the trigonometric identity for cotangent:

• Cotangent Identity: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Applying this understanding, we rewrite the fraction in terms of cotangent, leading to \( \ln|\cot \theta| \). This step not only simplifies the expression but also links back to fundamental trigonometric relationships, making the expression easier to solve or interpret in further mathematical contexts.