Problem 55 - Precalculus Essentials

Question

Derive the identity for \(\tan (\alpha+\beta)\) using $$\tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)}$$ After applying the formulas for sums of sines and cosines, divide the numerator and denominator by \(\cos \alpha \cos \beta\)

Step-by-Step Solution

Answer

The identity for \( \tan (\alpha+\beta) \) is \( \tan (\alpha+\beta)= \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

Step 1: Apply Sine Addition Formula

Start by applying the sum of angles formula to the sine function in the numerator of the fraction: \( \sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

Step 2: Apply Cosine Addition Formula:

Then, apply the sum of angle formula to the cosine function in the denominator of the fraction: \( \cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

Step 3: Divide by Αnd Substitution

After applying those formulas, divide both the numerator and the denominator by \( \cos \alpha \cos \beta \), and replace \( \sin \alpha / \cos \alpha \) with \( \tan \alpha \) and \( \sin \beta / \cos \beta \) with \( \tan \beta \). You'll get: \( \tan (\alpha+\beta)= \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

Key Concepts

Angle Sum Formulas

The angle sum formulas are essential tools in trigonometry, helping us break down complex angle expressions into simpler calculations. These formulas allow us to express the sine or cosine of the sum of two angles, which is extremely useful for solving various trigonometric problems. For instance, the sine addition formula is expressed as:

• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

This tells us that the sine of a sum of two angles is the sum of the products of the sine and cosine of each individual angle. It allows us to link angles and helps in transforming expressions into more manageable forms.
Similarly, the cosine addition formula relates to the cosine of a sum:

• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

This formula helps us see how the cosine of two added angles can be decomposed by subtracting the product of their sine functions from the product of their cosine functions. Understanding these formulas forms a foundation for tackling more intricate trigonometric identities and equations.

Tangent Addition Formula

The tangent addition formula is derived from the relationship between the sine and cosine addition formulas. This particular formula provides a way to calculate the tangent of the sum of two angles. Starting from:

• \( \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \)

By applying the angle sum formulas for sine and cosine, we can substitute these expressions accordingly. The division by \( \cos \alpha \cos \beta \) is a clever step that simplifies the expression, leading to the neat result for the tangent addition:

• \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

This formula is incredibly useful for simplifying complex trigonometric problems and often appears in calculus and physics. It underscores how angle relationships are interconnected and can be expressed elegantly.

Sine and Cosine Formulas

The sine and cosine addition formulas are cornerstones in trigonometry, but they extend beyond the angle addition to various transformations. These formulas not only help in simplifying sums like in the angle addition formulas, but they also pave the way for understanding other key identities, such as the double angle or half angle identities.

• Sine Addition Formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Cosine Addition Formula: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

While these focus on angle sums, similar rules apply for angle differences, enhancing their flexibility in multiple contexts. They help to understand how separate angles interact and influence overall trigonometric values. They offer an essential toolkit for all sorts of calculations, whether manipulating algebraic expressions or solving real-world measurement problems.