Problem 137 - Precalculus with Limits

Question

In Exercises 133 - 138, determine whether or not the equation is an identity, and give a reason for your answer. \( \sin \theta \csc \theta = 1 \)

Step-by-Step Solution

Answer

Yes, the equation \( \sin \theta \csc \theta = 1 \) is an identity because for any value of \( \theta \) (except where sin is 0) the left side simplifies to equal the right side.

Step 1: Definition of Cosecant

Firstly, remember that the cosecant of an angle is defined as \( \csc \theta = 1/ \sin \theta\). It’s the reciprocal of the sine function.

Step 2: Substitution

Substitute the definition of cosecant into the equation. The equation now reads \( \sin \theta *(1/ \sin \theta) = 1\).

Step 3: Simplifying the Equation

Simplify the equation. The sine function \( \sin \theta \) cancels out, resulting in \( 1 = 1\).

Key Concepts

Cosecant Function

When exploring the world of trigonometry, the cosecant function plays a crucial role, especially when delving into the relationships between different trigonometric functions. The cosecant, denoted as \( \text{csc} \theta \), is defined as the reciprocal of the sine function. This means that it can be expressed as \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \).
Understanding the cosecant function requires familiarity with the sine function. Imagine a right triangle, where \( \text{sin} \theta \) represents the ratio of the length of the opposite side to the hypotenuse. Conversely, the cosecant function represents the ratio of the length of the hypotenuse to the opposite side.
This function is essential in various fields of science and engineering, such as in wave mechanics or whenever the concept of periodicity and oscillation comes into play. Learning about the cosecant aids in comprehending concepts like wave amplitude and other phenomena that can be modeled trigonometrically.

Sine Function

The sine function, symbolized as \( \text{sin} \theta \), is one of the fundamental trigonometric functions and is pivotal to understanding various phenomena in mathematics. It describes the relationship between an angle of a right triangle and the ratios of its sides.
Specifically, the sine of an angle \( \theta \) in a right-angled triangle is the ratio of the length of the side opposite to \( \theta \) to the length of the hypotenuse. The sine function is defined for all real numbers and its values range between -1 and 1. It is an odd function, meaning that \( \text{sin}(-\theta) = -\text{sin}(\theta) \).
Applications of the sine function extend far beyond geometry, including physics, engineering, and even music theory, as it helps in the analysis of sound waves and harmonics. Moreover, understanding the sine function is crucial for grasping other trigonometric identities and concepts.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are an integral part of trigonometry, offering a different perspective on the relationships between the angles and sides of a triangle. The primary trigonometric functions – sine (sin), cosine (cos), and tangent (tan) – each have a corresponding reciprocal function: cosecant (csc), secant (sec), and cotangent (cot), respectively.
These reciprocal functions are defined as follows:

• Cosecant: \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \)
• Secant: \( \text{sec} \theta = \frac{1}{\text{cos} \theta} \)
• Cotangent: \( \text{cot} \theta = \frac{1}{\text{tan} \theta} \)

Learning about these functions is not just about understanding their definitions but also their applications. For instance, reciprocal functions can simplify complex trigonometric expressions or equations, making them easier to solve. They also play a vital role when working with trigonometric identities, which often involve expressing one function in terms of another to establish equivalencies or derive new formulas. Grasping these concepts is key to success in higher-level mathematics, physics, and engineering problems where trigonometry is applied.