Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use the addition formulas for \(\sin x\) and \(\cos x\) to derive the addition and subtraction formulas for \(\tan x\). $$ \begin{array}{l} \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} \\ \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} \end{array} $$

Step-by-Step Solution

Answer

The formulas for \( tan(A + B) \) and \( tan(A - B) \) are derived to be \( tan(A + B) = \frac{tan A + tan B}{1 - tan A tan B} \) and \( tan(A - B) = \frac{tan A - tan B}{1 + tan A tan B} \) respectively using the addition and subtraction formulas for sine and cosine.

Step 1: Applying Identity for Tangent

Substitute \( tan x = \frac{sin x}{cos x} \) into the formula. So, \( tan(A + B) = \frac{sin(A + B)}{cos(A + B)} \) and \( tan(A - B) = \frac{sin(A - B)}{cos(A - B)} \).

Step 2: Substituting Addition Formulas for Sine and Cosine

For \( tan(A + B) \), substitute \( sin(A + B) = sin A cos B + cos A sin B \) and \( cos(A + B) = cos A cos B - sin A sin B \) . Repeat the process for \( tan(A - B) \) , but remember that \( sin(A - B) = sin A cos B - cos A sin B \) and \( cos(A - B) = cos A cos B + sin A sin B \).

Step 3: Simplifying Complex Fractions

Divide the numerator and denominator of both \( tan(A + B) \) and \( tan(A - B) \) formulas with \( cos A cos B \) . This will result in the expression \( tan(A + B) = \frac{tan A + tan B}{1 - tan A tan B} \) and \( tan(A - B) = \frac{tan A - tan B}{1 + tan A tan B} \).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for simplifying and solving trigonometric expressions and equations. Identities are equations that hold true for all values of the variable within their domains. Among the most fundamental are the reciprocal identities, quotient identities, and pythagorean identities. The quotient identity for tangent is particularly relevant when dealing with addition or subtraction formulas: it is defined as \( \tan x = \frac{\sin x}{\cos x} \). This identity allows us to express the tangent function in terms of sine and cosine, paving the way for the derivation of more complex identities.

For example, to derive the tangent addition formula, one can start by expressing \( \tan(A+B) \) as \( \frac{\sin(A+B)}{\cos(A+B)} \) using the quotient identity. This approach reveals how interconnected these trigonometric functions are and highlights the importance of mastering the basic identities to progress in understanding trigonometry.

Sine and Cosine Addition Formulas

The addition formulas for sine and cosine are pivotal in extending trigonometric functions to compound angles. For any angles \( A \) and \( B \) these formulas are:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Similarly, we have subtraction formulas:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas are derived using the unit circle or by decomposing and reassembling the waves represented by sine and cosine. They allow us to combine two angles and find the sine or cosine of their sum or difference, which is an essential step in deriving the addition and subtraction formulas for the tangent function.

Tangent Function Properties

The tangent function exhibits several unique properties that are essential to understand when working with trigonometric expressions. Some of these are:

• Periodicity: \( \tan x \) is periodic with a period of \( \pi \) radians, meaning \( \tan (x + \pi) = \tan x \).
• Symmetry: \( \tan (-x) = -\tan x \) indicating the function is odd, symmetric about the origin.
• Asymptotes: Unlike sine and cosine, the tangent function has vertical asymptotes where it is undefined, which occur at \( \frac{\pi}{2} + n\pi \) for any integer \( n \).

These properties affect how the tangent function behaves and how it interacts with other trigonometric functions. When dealing with addition and subtraction problems, being aware of these can assist in predicting and understanding the outcome of combinations of angles.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a key skill in mathematics, often involving a mix of algebraic manipulation and strategic application of trigonometric identities. It's important to recognize patterns and common trigonometric identities to simplify complex expressions effectively. For instance, converting all functions to sines and cosines can often make it easier to see underlying relationships:

• To simplify \( \tan(A + B) = \frac{\sin(A+B)}{\cos(A+B)} \), we apply the addition formulas, then separate the terms, and finally divide by \( \cos A\cos B \) to obtain the simplified tangent addition formula.
• The same process with appropriate subtraction formulas leads to the tangent subtraction formula.

If you encounter a complex expression, breaking it down into smaller parts and using known identities will gradually simplify the expression to its simplest form. It's like solving a puzzle, with each identity being a piece that helps you see the full picture.