Problem 42 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\csc \theta+1}{\left(1 / \sin ^{2} \theta\right)+\csc \theta}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \sin \theta \).

Step 1: Recognize Trigonometric Identities

First, recall the trigonometric identity for cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). We will use this identity to rewrite the expression in terms of sine.

Step 2: Rewrite the Expression with Sine

Substitute \( \csc \theta = \frac{1}{\sin \theta} \) into both the numerator and the denominator of the expression: \( \frac{\frac{1}{\sin \theta} + 1}{\left(\frac{1}{\sin^2 \theta}\right) + \frac{1}{\sin \theta}} \).

Step 3: Simplify the Numerator

Simplify the numerator by combining the terms over a common denominator: \( \frac{\frac{1}{\sin \theta} + 1}{\text{...}} = \frac{\frac{1 + \sin \theta}{\sin \theta}}{\text{...}} \).

Step 4: Simplify the Denominator

Simplify the denominator by combining the terms over the common denominator \( \sin^2 \theta \):\( \left(\frac{1}{\sin^2 \theta}\right) + \frac{1}{\sin \theta} = \frac{1 + \sin \theta}{\sin^2 \theta} \).

Step 5: Divide Numerator by Denominator

Perform the division \( \frac{\frac{1 + \sin \theta}{\sin \theta}}{\frac{1 + \sin \theta}{\sin^2 \theta}} \). When dividing fractions, you multiply by the reciprocal:\[ \frac{1 + \sin \theta}{\sin \theta} \times \frac{\sin^2 \theta}{1 + \sin \theta} \].

Step 6: Cancel Common Terms

Cancel the common term \( 1 + \sin \theta \) in the numerator and denominator:\[ \frac{\sin^2 \theta}{\sin \theta} \].

Step 7: Simplify the Result

Simplify the expression by dividing \( \sin^2 \theta \) by \( \sin \theta \), which results in \( \sin \theta \).

Key Concepts

Sine Function

In trigonometry, the sine function is one of the primary trigonometric functions, which relates the angles and sides of right triangles. It is denoted as \( \sin \theta \), where \( \theta \) is the angle in question. In the context of a unit circle, the sine of an angle is equivalent to the y-coordinate of the point where the terminal side of the angle intersects the circle.

• The sine function is periodic, with a period of \( 2\pi \) radians or 360 degrees.
• Its range is between -1 and 1, inclusive.
• The sine of an angle in a right triangle can be found as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Understanding the sine function is essential when simplifying expressions involving trigonometric identities, as it appears frequently across various identities and equations.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function. Specifically, \( \csc \theta = \frac{1}{\sin \theta} \). Because it is derived from the sine function, it is important to grasp how it behaves and relates to the sine.

• Since \( \sin \theta \) can range from -1 to 1, \( \csc \theta \) can take any value except those within the range -1 and 1.
• The cosecant function is undefined whenever the sine function is zero, as in these cases, you would be dividing by zero.
• Like the sine function, the cosecant is periodic, repeating every \( 2\pi \) radians.

In trigonometric identities and expressions, substituting \( \csc \theta \) with the sine's reciprocal can significantly aid in simplifying complex expressions.

Simplifying Expressions

Simplifying trigonometric expressions involves reducing them to their simplest form. This process often requires the use of trigonometric identities and algebraic manipulation.

• Recognize and apply fundamental identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \), to make simplifications.
• Convert complicated terms to simpler forms using substitutions, such as replacing \( \csc \theta \) with \( \frac{1}{\sin \theta} \).
• Look for common factors or terms that can be cancelled out to streamline the expression.

The goal is to make the expression easier to evaluate or to prepare it for further manipulation. Guidance in simplifying involves recognizing patterns and strategically applying known identities.

Algebraic Manipulation

Algebraic manipulation in trigonometry involves using basic algebraic operations to transform expressions. This process is crucial when working with trigonometric identities to simplify or solve problems.

• Combining like terms is often necessary, especially when expressions include fractions, to achieve a single simplified term.
• Using the reciprocal or multiplication and division, particularly with fractions, helps adjust and simplify complex expressions. For instance, in dividing fractions, you multiply by the reciprocal.
• Cancellation of terms, when possible, is a straightforward method to reduce expressions to their most basic form, as seen in the final steps of our solution, where common terms are cancelled before final simplification.

Proficiency in these techniques allows smooth navigation through challenging trigonometric problems, making them more manageable and less intimidating.