Problem 253 - Precalculus

Question

For the following exercises, suppose \(\sin t=\frac{x}{x+1}\) $$ \csc t $$

Step-by-Step Solution

Answer

\( \csc t = \frac{x+1}{x} \)

Step 1: Understanding \\( \\csc t \\\)

The function \( \csc t \) is the cosecant of \( \ t\ \), which is the reciprocal of the sine of \( \ t\ \). So, \( \csc t = \frac{1}{\sin t}\).

Step 2: Substituting \\( \sin t\\ \)

Given that \( \sin t = \frac{x}{x+1}\ \), substitute this expression into the cosecant. Thus, \[ \csc t = \frac{1}{\frac{x}{x+1}} \]

Step 3: Simplifying the Expression

To simplify the expression \( \frac{1}{\frac{x}{x+1}}\ \), we take the reciprocal of the fraction. This gives us \[ \csc t = \frac{x+1}{x} \]

Step 4: Finalizing the Solution

Thus, the expression for \( \csc t\ \) in terms of \( \ x\ \) is \( \frac{x+1}{x}\ \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc t \), plays an important role in trigonometry. It is one of the six fundamental trigonometric functions. This function is defined as the reciprocal of the sine function. That means for a given angle \( t \), the cosecant is calculated as \( \csc t = \frac{1}{\sin t} \). This is essential knowledge when you are working with trigonometric identities and calculating the ratios of sides in right triangles.
Understanding the cosecant function deepens your comprehension of how different trigonometric functions relate to each other. While it might seem complex at first, the simplicity lies in its reciprocal relationship to the sine function, making it straightforward to compute once you have the sine value.

Reciprocal Identities

Reciprocal identities are crucial in trigonometry because they help redefine one function in terms of another. For instance, the cosecant function uses the reciprocal identity \( \csc t = \frac{1}{\sin t} \). Similarly, other trigonometric functions also have reciprocal identities:

• \( \sec t = \frac{1}{\cos t} \)
• \( \cot t = \frac{1}{\tan t} \)

These identities allow you to switch between different ways of representing trigonometric functions, which can be particularly useful when simplifying expressions or solving equations. Being familiar with reciprocal identities can significantly help in transforming complex trigonometric expressions into simpler forms.

Simplifying Expressions

Simplifying expressions is a fundamental skill in trigonometry. It involves reducing an expression to its simplest form. The process often requires using identities, such as reciprocal identities, to make the expression easier to understand or work with.
In the given problem, simplifying \( \frac{1}{\frac{x}{x+1}} \) was vital. We applied the reciprocal of a fraction rule, turning the nested fraction into \( \frac{x+1}{x} \). This simplification made the expression clear and straightforward.
When simplifying expressions, always look for opportunities to use basic arithmetic and algebraic rules, such as the reciprocal, to reduce complexity. Practice with simplifying diverse expressions will enhance your ability to see these opportunities quickly.