Problem 35 - Precalculus: Functions and Graphs

Question

Use the Pythagorean Identities to write the expression as an integer. (a) \(\tan ^{2} 4 \beta-\sec ^{2} 4 \beta\) (b) \(4 \tan ^{2} \beta-4 \sec ^{2} \beta\)

Step-by-Step Solution

Answer

(a) -1, (b) -4.

Step 1: Recall Pythagorean Identity

The Pythagorean Identity relating tangent and secant is: \(1 + \tan^2 \theta = \sec^2 \theta\). We will use this identity to simplify both expressions (a) and (b).

Step 2: Simplify Expression (a) Using Identity

For expression (a), \(\tan^2(4\beta) - \sec^2(4\beta)\), we apply the identity. Substitute \(\sec^2(4\beta) = 1 + \tan^2(4\beta)\). Then, the expression becomes \(\tan^2(4\beta) - (1 + \tan^2(4\beta)) = \tan^2(4\beta) - \tan^2(4\beta) - 1 = -1\).

Step 3: Simplify Expression (b) Using Identity

For expression (b), \(4\tan^2(\beta) - 4\sec^2(\beta)\), we also use the identity \(\sec^2(\beta) = 1 + \tan^2(\beta)\). Substitute this into the expression to get \(4\tan^2(\beta) - 4(1 + \tan^2(\beta)) = 4\tan^2(\beta) - 4 - 4\tan^2(\beta)\). Simplifying gives \(-4\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true for any angle. These identities are essential because they help simplify trigonometric expressions and solve equations. One of the most important trigonometric identities is the Pythagorean Identity. This identity connects the sine, cosine, and tangent functions in a simple equation.

The basic Pythagorean Identity is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

From this, we derive the identity that involves tangent and secant:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)

This identity is incredibly useful, especially when simplifying or manipulating expressions involving tangent and secant functions. By understanding and using this identity, you can easily transition between these trigonometric functions to find simpler forms or specific values.

Tangent and Secant Functions

The tangent and secant functions are trigonometric functions that are derived from the fundamental sine and cosine functions. Understanding these functions and their properties is key to solving a variety of trigonometric problems.

The tangent function, denoted as \( \tan(\theta) \), represents the ratio of the sine of an angle to the cosine of that angle:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

Tangent is periodic with a period of \( \pi \) radians, meaning it repeats its values every \( \pi \) radians.

The secant function, denoted as \( \sec(\theta) \), is the reciprocal of the cosine function:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

Secant is also periodic, repeating every \( 2\pi \) radians, just like the cosine function. Using these relationships, we can manipulate expressions involving tangent and secant to achieve simpler forms, especially using the identity \( 1 + \tan^2 \theta = \sec^2 \theta \), which allows transformation between the two.

Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics that involves rewriting expressions in the simplest form possible. For trigonometric expressions, this often means minimizing the number of terms or converting them to more basic trigonometric identities.

When simplifying expressions, it’s essential to identify opportunities to use known identities or mathematical properties. For trigonometric expressions like the ones given, you will primarily focus on using the Pythagorean Identity:

• Substitute expressions using identities like \( 1 + \tan^2 \theta = \sec^2 \theta \).
• Cancel out similar terms.
• Simplify constants or coefficients.

For example, in the expression \( \tan^2(4\beta) - \sec^2(4\beta) \), using the identity simplifies it directly to a constant \(-1\). Similarly, multiplying every term by a common factor, like \(4\) in the expression \(4\tan^2(\beta) - 4\sec^2(\beta)\), keeps the structure intact but does not change the form, resulting in a simple \(-4\). This problem solving technique significantly reduces complexity and helps in finding direct integer values from trigonometric expressions.