Problem 36 - Contemporary Precalculus

Question

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(\sin t+\csc t)\left(\sin ^{2} t+\csc ^{2} t-1\right)$$

Step-by-Step Solution

Answer

Question: Simplify the expression ($$ (\sin t+\csc t)\left(\sin ^{2} t+\csc ^{2} t-1\right) $$). Answer: $$ \sin^3 t - 1 + \frac{1}{\sin t} $$

Step 1: Recall the definitions and identities

Recall the following definitions and identities: 1. csc(t) = 1/sin(t) 2. sin^2(t) + cos^2(t) = 1

Step 2: Perform the multiplication

Let's multiply the two given expressions: ($$ (\sin t+\csc t)\left(\sin ^{2} t+\csc ^{2} t-1\right) $$) By the distributive property, we can multiply each term in the first expression with each term in the second expression. $$ (\sin t \cdot \sin^2 t) + (\sin t \cdot \csc^2 t) - (\sin t) + (\csc t \cdot \sin^2 t) + (\csc t \cdot \csc^2 t) - (\csc t) $$

Step 3: Substitute the definitions and identities

Now, let's replace csc(t) with 1/sin(t) and rearrange the terms: $$ (\sin t \cdot \sin^2 t) - (\sin t) + (\frac{1}{\sin t} \cdot \sin^2 t) - (\frac{1}{\sin t}) + (\frac{1}{\sin t} \cdot \frac{1}{\sin^2 t}) $$

Step 4: Simplify the expression

Now, let's simplify each term: $$ \sin^3 t - \sin t + \sin t - 1 + \frac{1}{\sin t} $$ Notice that ('\(- \sin t\)' and '\(+ \sin t\)') cancel each other out. We are left with: $$ \sin^3 t - 1 + \frac{1}{\sin t} $$ This is the simplified expression for the original problem.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially in dealing with angles and periodic phenomena. They relate angles in a right triangle to the ratio of its sides. Here are some of the basic trigonometric functions involved:

• Sine (\( \sin \)): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. It is often written as \( \sin(t) \).
• Cosecant (\( \csc \)): Cosecant is the reciprocal of sine. If \( \sin(t) \) is the opposite side over the hypotenuse, \( \csc(t) \) is the hypotenuse over the opposite side, or \( \csc(t) = \frac{1}{\sin(t)} \).

These functions are interconnected by various identities that help simplify complex expressions. Understanding these identities allows us to manipulate and solve trigonometric equations more efficiently.

Simplifying Expressions

Simplifying expressions involves combining like terms and applying algebraic identities to reduce an expression to its simplest form. In trigonometry, this often involves using known identities to merge terms. Consider an expression like:

• \( \sin^2(t) + \csc^2(t) - 1 \)

Knowing that \( \csc(t) = \frac{1}{\sin(t)} \), substitution can occur to express everything in terms of \( \sin \). This step reduces complexity, allowing us to see potential cancellations or simplifications. After substitution and combining similar terms, expressions often become more manageable.

Mathematical Operations

Mathematical operations include the basic processes of addition, subtraction, multiplication, and division. In the context of our original exercise, multiplication takes center stage. We explore the operation by multiplying:

• \( (\sin t + \csc t) \) with \( (\sin^2 t + \csc^2 t - 1) \)

Each term in the first parentheses multiplies every term in the second. This type of multiplication makes use of the distributive property, expanding the expression into multiple terms. Each operation step brings us closer to the simplified form by highlighting terms that can be combined or cancelled later. Mastery of these operations is crucial for solving complex trigonometric equations.

Distributive Property

The distributive property is a fundamental concept in algebra. It lets you multiply a single term by several terms inside parentheses and distribute it to each term individually. In the exercise, the distributive property helps to expand:

• \( (\sin t + \csc t) \cdot (\sin^2 t + \csc^2 t - 1) \)

Apply the distributive property by multiplying \( \sin t \) and \( \csc t \) with each component inside the second parentheses. This results in terms like \( \sin t \cdot \sin^2 t \) and \( \csc t \cdot \csc^2 t \). Handling each product individually helps identify simplifying opportunities later. The property is key to breaking down expressions into a form that's easier to analyze and simplify.