Problem 36 - Precalculus: Functions and Graphs

Question

Use the Pythagorean Identities to write the expression as an integer. (a) \(\csc ^{2} 3 \alpha-\cot ^{2} 3 \alpha\) (b) \(3 \csc ^{2} \alpha-3 \cot ^{2} \alpha\)

Step-by-Step Solution

Answer

(a) 1 (b) 3

Step 1: Recall the Pythagorean Identity

One of the Pythagorean identities states that for any angle \(x\), the identity \( \csc^2 x - \cot^2 x = 1 \) holds true.

Step 2: Apply the Identity to Part (a)

For the expression \( \csc^2 (3\alpha) - \cot^2 (3\alpha) \), apply the identity. Replace \( \csc^2 (3\alpha) - \cot^2 (3\alpha) \) with 1, since \( \csc^2 x - \cot^2 x = 1 \) for any angle \(x\). Thus, \( \csc^2 (3\alpha) - \cot^2 (3\alpha) = 1 \).

Step 3: Apply the Identity to Part (b)

For the expression \( 3 \csc^2 \alpha - 3 \cot^2 \alpha \), factor out the 3, resulting in \( 3(\csc^2 \alpha - \cot^2 \alpha) \). By applying the identity \( \csc^2 x - \cot^2 x = 1 \), simplify to \( 3 \times 1 = 3 \).

Key Concepts

Understanding Cosecant and Cotangent

Cosecant and cotangent are trigonometric functions that are crucial in understanding certain mathematical concepts, including Pythagorean identities. Let's explore these functions more closely.

Cosecant, denoted as \( \ ext{csc} \), is the reciprocal of the sine function. That means:

• \( \ ext{csc}(x) = \frac{1}{\sin(x)} \)

It is only defined for angles where the sine is not zero.

Cotangent, represented by \( \text{cot} \), is another trigonometric ratio, formulated as:

• \( \text{cot}(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \)

It is valid where sine is not zero, as it involves division by sine as well.
These definitions establish cosecant and cotangent as integral parts of solving trigonometric problems and when dealing with angles represented in identities.

Exploring Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the involved variable. One key identity is the Pythagorean identity, which is the foundation for many trigonometric simplifications.

The Pythagorean identity for cosecant and cotangent states:

• \( \csc^2(x) - \cot^2(x) = 1 \)

This identity tells us that the squared cosecant minus the squared cotangent always equals one.

These identities are powerful tools for simplifying complex trigonometric expressions, transforming them into much simpler forms. For instance, when faced with the expression \( \ ext{csc}^2(3\alpha) - \ ext{cot}^2(3\alpha) \), substituting from the identity immediately gives you 1. This simplification shows how effectively trigonometric identities can be used to reduce problem complexity.

The Art of Integer Simplification

Integer simplification is an essential skill in math, requiring careful handling of expressions to transform them into simpler or more comprehendible forms. This often involves factoring, applying identities, or other algebraic techniques.

Consider this example: to simplify \( 3 \ ext{csc}^2(\alpha) - 3 \ ext{cot}^2(\alpha) \), you first recognize that both terms are multiplied by the coefficient 3.

By applying the Pythagorean identity \( \ ext{csc}^2(x) - \ ext{cot}^2(x) = 1 \), you can rewrite the expression as:

• \( 3(\csc^2(\alpha) - \cot^2(\alpha)) \)

Substituting from the identity turns this into \( 3 imes 1 = 3 \). The final step involves realizing that the expression simplifies to just the integer 3.

This process highlights the logical steps involved in simplifying expressions, showing how an initial complex form can be reduced to something straightforward.