Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\log \csc \theta=-\log \sin \theta$$

Step-by-Step Solution

Answer

The identity \(\log \csc \theta = -\log \sin \theta\) is verified by using logarithmic properties of reciprocals.

Step 1: Recall the Identity of Cosecant

The cosecant function, \(\csc \theta\), is defined as the reciprocal of the sine function. Therefore,\[\csc \theta = \frac{1}{\sin \theta}.\]

Step 2: Apply Logarithm Properties

We need to apply the properties of logarithms to transform \(\log \csc \theta\). Using the change of base formula:\[\log \csc \theta = \log \left( \frac{1}{\sin \theta} \right).\]By the property of logarithms, \(\log \left( \frac{1}{a} \right) = -\log a\), we can simplify:\[\log \left( \frac{1}{\sin \theta} \right) = -\log \sin \theta.\]

Step 3: Verify the Transformed Expression

After transforming \(\log \csc \theta\) using logarithmic identities, we find that:\[\log \csc \theta = -\log \sin \theta.\] This matches the right-hand side of the given identity.

Key Concepts

Understanding Cosecant

The cosecant function, denoted as \( \csc \theta \), might not be as familiar to students as sine or cosine, yet it's equally important. It is defined as the reciprocal of the sine function. This means for any angle \( \theta \), you can find the cosecant by taking one divided by the sine of that angle:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This relationship comes in handy when working through various trigonometric identities and equations. By understanding this reciprocal connection, you can easily switch between sine and cosecant, simplifying complex expressions. In contexts like the given exercise, knowing how to express cosecant in terms of sine is foundational for using other mathematical tools, such as logarithms, effectively.

Exploring Logarithmic Properties

Logarithms and their properties make it possible to transform complex multiplicative relationships into more manageable additive ones. In the context of verifying the exercise's identity, we leverage several key properties:

• Firstly, \( \log \left( \frac{1}{a} \right) = -\log a \). This property shows how the logarithm of a reciprocal turns into a negative logarithm.


• Secondly, the change of base formula is often used to simplify expressions further.

By applying these properties, we can see how the logarithm of cosecant simplifies to \( -\log \sin \theta \).
The ability to manipulate logarithmic expressions using these rules is essential. It allows us to break down the problem and transform \( \log \csc \theta \) into a familiar form that matches the desired equation.

Diving into the Sine Function

The sine function is a fundamental building block in trigonometry, central to understanding various trigonometric identities and transformations. Defined traditionally on a unit circle, sine relates to the y-coordinate of a point corresponding to a given angle \( \theta \):

• \( \sin \theta \) captures these relationships and oscillates between -1 and 1 as \( \theta \) varies.


• This periodic nature of sine, with its regular cycles and behavior, forms the basis for reciprocal identities like cosecant.

Understanding \( \sin \theta \) is crucial not only in mathematical reasoning but also in applied fields such as physics and engineering. With its cyclical pattern, the sine function serves as a predictable model for waves and oscillations, illustrating its scope beyond pure math. In our identity verification task, the sine function’s role is pivotal; by expressing cosecant as \( \frac{1}{\sin \theta} \), the reciprocal relationship helps to bridge the equation to logarithms, making transformations seamless and coherent.