Problem 33 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\frac{\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta}=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}$$

Step-by-Step Solution

Answer

The given identity is verified, both sides equal \(\tan(\alpha + \beta)\).

Step 1: Recognize Trigonometric Identities

To verify the identity, recall two important trigonometric identities:1. **Sine of a sum formula:** \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)2. **Cosine of a sum formula:** \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)

Step 2: Substitute Trigonometric Identities

Substitute the identities from Step 1 into the left-hand side (LHS) of the given equation:\[\frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)}\]Thus, the LHS simplifies to \(\tan(\alpha + \beta)\) since \(\frac{\sin(\text{angle})}{\cos(\text{angle})} = \tan(\text{angle})\).

Step 3: Simplify Right-Hand Side

The right-hand side (RHS) \(\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\) is the tangent addition formula:\(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\)

Step 4: Conclude Identity

Since both the LHS and RHS simplify to \(\tan(\alpha + \beta)\), the identity is verified. Therefore:\(\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\) holds true.

Key Concepts

Sine of a Sum Formula

The Sine of a Sum Formula is an essential part of understanding the behavior of trigonometric functions when combining angles. It states:

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

This formula helps us find the sine of the sum of two angles \( \alpha \) and \( \beta \). By decomposing the sin of the sum into a sum of products, it facilitates easier computation for certain types of problems.
The formula describes how two separate angles contribute to the resultant sine value. Each term in the formula represents how the sine and cosine components of those angles interact. There can be practical uses when working with wave mechanics or when analyzing periodic phenomena.

Cosine of a Sum Formula

The Cosine of a Sum Formula is another fundamental identity in trigonometry. It is used to calculate the cosine of the sum of two angles. The formula is:

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

Unlike sine, the cosine addition has a subtraction between the sine components of the angles. The identity helps expand expressions into a more usable form, often making it easier to solve complex trigonometric problems.
The cosine of a sum formula is not just a mathematical tool, but also explains many natural phenomena, such as light waves interacting or the interference patterns seen in physics. In our goal to verify trigonometric identities, this formula is pivotal as it aids in transforming expressions to explore equivalences, like those seen in the verification of the exercise identity.

Tangent Addition Formula

The Tangent Addition Formula provides a way to compute the tangent of the sum of two angles. It is given by:

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

This particular formula is useful for simplifying expressions where the sum of two angles' tangents is involved. Notice how the formula makes use of both the sum and product of individual tangents.
The utility of this identity lies in its ability to break down a potentially complex operation (the tangent of a sum) into simpler, manageable parts. Understanding and using such a formula is important especially when studying precalculus or calculus. The formula is indispensable in verifying identities, solving equations, and modeling scenarios where angles are interacting, demonstrating its broad applicability.