Problem 15 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \((\csc \theta)(\sin \theta)\)

Step-by-Step Solution

Answer

The simplest form of the expression is 1.

Step 1: Write in terms of sine and cosine

The expression is \((\csc \theta)(\sin \theta)\). We know that \(\csc \theta\) is the reciprocal of \(\sin \theta\). Therefore, \(\csc \theta = \frac{1}{\sin \theta}\). Now rewrite the expression using this identity: \(\frac{1}{\sin \theta} \times \sin \theta\).

Step 2: Simplify the expression

Now simplify the expression \(\frac{1}{\sin \theta} \times \sin \theta\). Here, \(\sin \theta\) in the numerator and denominator cancel each other out. Therefore, you are left with the expression \(1\).

Key Concepts

Sine and Cosine Relationship

A fundamental understanding in trigonometry is the relationship between the sine and cosine functions. These two functions form the foundation of the unit circle. In the unit circle, each point is represented as \(cos \theta, sin \theta\), linking angle measurements to the horizontal and vertical components of the circle.

• Sine function (\(sin \theta\)): Represents the y-coordinate or vertical component of the angle on the unit circle.
• Cosine function (\(cos \theta\)): Represents the x-coordinate or horizontal component of the angle on the unit circle.

These functions are interconnected through several key identities, which help simplify complex trigonometric expressions.

• One fundamental identity is the Pythagorean identity: \(sin^2 \theta + cos^2 \theta = 1\).
• This identity stems from the Pythagorean Theorem applied to the unit circle.

The reciprocal and quotient identities also highlight the connection between sine and cosine:

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

These relationships are crucial for simplifying expressions and solving trigonometric equations.

Cosecant Function

The cosecant function, often written as \(csc \theta\), is the reciprocal of the sine function. It is defined as \(csc \theta = \frac{1}{sin \theta}\). This function is particularly useful in the analysis and simplification of trigonometric identities and expressions.

• Understanding \(csc \theta\): Whenever \(sin \theta\) is involved in an expression, \(csc \theta\) can replace or interact with it in a reciprocal relationship.
• Range and Domain: \csc \theta\ is undefined whenever \(sin \theta = 0\). On the unit circle, this equates to angles where \(\theta = n\pi\) (where \(n\) is an integer), i.e., at 0, \(\pi\), 2\(\pi\), etc.

In expression simplification, replacing \(csc \theta\) with \(\frac{1}{sin \theta}\) allows us to work algebraically with other trigonometric identities:

• For instance, in the expression \( (csc \theta)(sin \theta)\), rewriting it using \(\frac{1}{sin \theta}\) simplifies the equation to 1 by eliminating \(sin \theta\).

The \(csc \theta\) function's role as a reciprocal enhances our ability to maneuver through trigonometric transformations and expression simplifications.

Expression Simplification

Expression simplification in trigonometry involves applying trigonometric identities to reduce complex expressions to simpler or more computationally efficient forms. A primary goal is to convert expressions into basic functions like sine and cosine.

• Step-by-step simplification:
• Identify any trigonometric identities or reciprocals within the expression. For example, notice that \(csc \theta = \frac{1}{sin \theta}\).
• Apply these identities: Reposers the trigonometric functions as their sine and cosine equivalents.
• Remove redundancy: Cancel common terms in the fraction, such as \(sin \theta\) in \(\frac{1}{sin \theta} \times sin \theta\).

In our original exercise, the expression \( (csc \theta)(sin \theta)\) uses the reciprocal identity. Begin by translating \(csc \theta\) to \(\frac{1}{sin \theta}\), simplifying the product to just 1 due to the \(sin \theta\) terms cancelling each other. This demonstrates the importance of recognizing reciprocal identities and cancelling terms in simplifying trigonometric expressions.