Problem 52 - Precalculus: Functions and Graphs

Question

Derive the subtraction formula for the tangent function.

Step-by-Step Solution

Answer

The subtraction formula for tangent is \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \).

Step 1: Understanding the Problem

The goal is to derive an expression for \( \tan(a - b) \) using the known tangent addition formula and trigonometric identities.

Step 2: Recall the Tangent Addition Formula

First, recall the formula for the tangent of a sum: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). This formula will be helpful in deriving the subtraction formula.

Step 3: Algebraic Substitution for Tangent Subtraction

To find \( \tan(a - b) \), consider it as \( \tan(a + (-b)) \). Using the addition formula, \( \tan(a - b) = \frac{\tan a + \tan(-b)}{1 - \tan a \tan(-b)} \). Since \( \tan(-b) = -\tan b \), substitute to get \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \).

Step 4: Verify the Derived Formula

Check the derived subtraction formula by considering special values, such as \( a = b \). The formula should simplify correctly, ensuring the derivation is accurate. In this case, if \( a = b \), \( \tan(a - b) = \tan(0) = 0 \), which matches the formula, \( \frac{\tan a - \tan a}{1 + \tan a \tan a} = 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are vital tools in mathematics, particularly when dealing with angles and their respective trigonometric functions. These identities are equations that are true for all values of the variable where both sides of the equation are defined. One of the most familiar trigonometric identities is the Pythagorean identity: \\[ \sin^2 \theta + \cos^2 \theta = 1 \].

Bootstrapping on these identities allows mathematicians to derive other important relationships used in various applications, such as physics and engineering. In deriving the tangent subtraction formula, the identity for the tangent of the negated angle, \( \tan(-b) = -\tan b \), is particularly useful. It illustrates how the tangent function is odd, meaning it reflects symmetrically over the origin in the Cartesian coordinate system.

Understanding these identities can simplify complex expressions and solve problems involving angles effectively. Mastery of trigonometric identities is a foundation skill for pursuing advanced topics in mathematics and related fields.

Tangent Addition Formula

The tangent addition formula is a crucial trigonometric identity that relates the tangent of the sum of two angles to the tangents of the individual angles. This formula is expressed as: \\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \].

Let's break down how this formula is useful. It allows us to determine the tangent of a summed angle, provided we know the tangents of the component angles, \( \tan a \) and \( \tan b \). This approach is particularly beneficial in problems where directly measuring or calculating an angle is challenging.

The formula results from the identities associated with sine and cosine: \\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] and \\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \].
By expressing tangent in terms of sine and cosine, we can derive the addition formula through a combination of algebraic manipulation and trigonometric identity application. Recognizing this formula enhances one's ability to solve trigonometric problems both quickly and correctly.

Algebraic Substitution

Algebraic substitution is a powerful method used to simplify expressions and derive new formulas. In mathematics, this involves replacing one part of an expression with another equivalent expression. This technique is pivotal in deriving the tangent subtraction formula.

To find \( \tan(a - b) \), we recognize it can be viewed as \( \tan(a + (-b)) \). This insight allows us to use the tangent addition formula by replacing \( b \) with \( -b \). Doing so, we substitute into the formula: \\[ \tan(a + (-b)) = \frac{\tan a + \tan(-b)}{1 - \tan a \tan(-b)}. \]
Since \( \tan(-b) = -\tan b \), substitution yields: \\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}. \]

This step showcases algebraic substitution's role in leveraging known identities to develop new ones. It demonstrates how understanding one concept deeply can lead to the discovery or derivation of another. Mastery of this technique simplifies problem-solving across various areas such as calculus, physics, and beyond. Hence, algebraic substitution is a key skill to learn and practice.