Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( 2 \sin \theta - 1 \).

Step 1: Identify Trigonometric Identities

Recall the definition of cosecant, where \( \csc \theta = \frac{1}{\sin \theta} \). This can help us simplify expressions involving \( \csc \theta \).

Step 2: Simplify Denominator

Rewrite the denominator \( \left(\frac{1}{\sin^{2}\theta}\right) + \csc \theta \) using the identity for \( \csc \theta \). It becomes \( \frac{1}{\sin^2 \theta} + \frac{1}{\sin \theta} \). By finding a common denominator, we have \( \frac{1 + \sin \theta}{\sin^2 \theta} \).

Step 3: Rephrase the Expression

Replace the denominator in the original expression with our simplified form from Step 2, giving us, \( \frac{\csc \theta + 1}{\frac{1 + \sin \theta}{\sin^2 \theta}} \).

Step 4: Simplify the Fraction

Flip the denominator fraction and multiply: \( (\csc \theta + 1) \times \frac{\sin^2 \theta}{1 + \sin \theta} \). Substitute back \( \csc \theta = \frac{1}{\sin \theta} \), giving: \[ \left(\frac{1}{\sin \theta} + 1\right) \times \frac{\sin^2 \theta}{1 + \sin \theta} \].

Step 5: Simplify the Product

Distribute \( \sin^2 \theta \) in the product: \( \frac{\sin^2 \theta}{\sin \theta} + \frac{\sin^2 \theta}{1 + \sin \theta} \). Simplify each term: \( \sin \theta + \frac{\sin^2 \theta}{1 + \sin \theta} \).

Step 6: Simplify Entire Expression

The second term \( \frac{\sin^2 \theta}{1 + \sin \theta} \) can be expressed as \( \sin \theta - 1 + 1 \), hence combining with the first term: \[ \sin \theta + \sin \theta - 1 \]. This gives: \( 2\sin \theta - 1 \).

Step 7: Conclude Simplification

Reassess and finalize: since no further simplification is apparent and steps verified, the simplified form of the expression is \( 2 \sin \theta - 1 \).

Key Concepts

Cosecant

To understand trigonometric identities, you must first know the basic functions, such as the cosecant. The cosecant, denoted as \( \csc \theta \), is a trigonometric function. It's related closely to the sine function. Specifically, it's the reciprocal of sine. Here's how we define it:

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

In simple terms, the cosecant function takes the value that sine gives you and flips it over a fraction. If sine gives 0.5, cosecant gives the inverse, or \( 2 \). Understanding this relationship can help break down trigonometric expressions, as seen in the simplification example. Knowing and applying the definition of cosecant allows us to transform expressions for further simplification.

Simplification of Expressions

Simplifying mathematical expressions means reducing them to their simplest form. This process is essential in solving trigonometric problems. In the given exercise, we started by identifying the relevant identities, such as the cosecant's relationship with sine.

Simplification usually involves:

• Identifying common factors or terms.
• Using mathematical identities to replace one form with an equivalent but simpler form.
• Combining like terms and reducing fractions.

In this particular expression, we observed how recognizing the identity associated with \( \csc \theta \) paved the way for finding a common denominator. This skill of transforming parts of the expression helps in simplifying the overall problem. In conclusion, simplification is all about making the expression easier to work with while maintaining equality.

Trigonometric Functions

Trigonometric functions form the foundation of trigonometry. They are functions of an angle and are crucial in understanding periodic phenomena. The main six functions include sine, cosine, tangent, secant, cosecant, and cotangent. Here's a quick recap:

• Sine (\( \sin \theta \)): Gives the y-coordinate on the unit circle.
• Cosine (\( \cos \theta \)): Provides the x-coordinate on the unit circle.
• Tangent (\( \tan \theta \)): Defined as \( \frac{\sin \theta}{\cos \theta} \).
• Secant (\( \sec \theta \)): The reciprocal of cosine, \( \frac{1}{\cos \theta} \).
• Cosecant (\( \csc \theta \)): The reciprocal of sine, \( \frac{1}{\sin \theta} \).
• Cotangent (\( \cot \theta \)): The reciprocal of tangent, \( \frac{1}{\tan \theta} \).

Each function has its unique properties and graphs, which are useful for modeling cyclical patterns such as sound waves, light waves, and tides. Understanding these functions deeply will enhance your ability to simplify and solve trigonometric expressions.

Trigonometric Fractions

Trigonometric fractions involve divisions of trigonometric expressions and often appear daunting. However, with practiced methods, such as finding common denominators and applying reciprocal identities, they can be simplified.

In the exercise, we faced a fraction containing cosecants and sine powers. The process began by adjusting the denominator using a common denominator, simplifying the entire expression successfully.

When dealing with trigonometric fractions:

• Look for opportunities to use identities like \( \csc \theta = \frac{1}{\sin \theta} \) to replace terms.
• Simplify complex fractions by flipping and multiplying.
• Ensure the final expression has the simplest achievable terms using arithmetic operations.

Mastering these techniques is excellent for those looking to solve advanced trigonometric expressions accurately and efficiently.