Problem 47 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}=\frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta} $$

Step-by-Step Solution

Answer

The given trigonometric identity \( \frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)} = \frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta} \) gets successfully verified after properly applying the sum and difference identities for \(\sin\) and \(\tan\), followed by simplification. The left-hand side (LHS) of the equation equals the right-hand side (RHS)

Step 1: Write down the given identity

The given identity is \( \frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)} = \frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta} \)

Step 2: Substitute the sum and difference identities for \(\sin\)

The sum and difference identities for \(\sin\) are \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \) and \(\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \). Substituting these into the given identity gives: \( \frac{\sin\alpha \cos\beta + \cos\alpha \sin\beta}{\sin\alpha \cos\beta - \cos\alpha \sin\beta} = \frac{\tan\alpha + \tan\beta}{\tan\alpha - \tan\beta} \)

Step 3: Substitute the definition for \(\tan\)

The definition for \(\tan\) is \(\tan x = \frac{\sin x}{\cos x}\). Substituting this into the RHS of the equation gives: \( \frac{\sin\alpha \cos\beta + \cos\alpha \sin\beta}{\sin\alpha \cos\beta - \cos\alpha \sin\beta} = \frac{\frac{\sin\alpha}{\cos\alpha} + \frac{\sin\beta}{\cos\beta}}{\frac{\sin\alpha}{\cos\alpha} - \frac{\sin\beta}{\cos\beta}} \)

Step 4: Simplify the right-hand side

After performing the simplification, the equation becomes \( \frac{\sin\alpha \cos\beta + \cos\alpha \sin\beta}{\sin\alpha \cos\beta - \cos\alpha \sin\beta} = \frac{\sin\alpha \cos\beta + \cos\alpha \sin\beta}{\sin\alpha \cos\beta - \cos\alpha \sin\beta} \)

Step 5: Verify the identity

Since both sides of the equation are equal, we can conclude that the given identity is verified

Key Concepts

Sum and Difference Identities

Sum and difference identities are major tools in trigonometry that help simplify and solve trigonometric expressions. They express the trigonometric functions of sums or differences of angles in terms of the trigonometric functions of individual angles.

These identities help unravel complex expressions into more manageable parts, which is particularly helpful in verifying identities and solving equations. Let's consider the sum and difference identities for the sine function, which forms part of our problem:

• Sum identity for sine: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Difference identity for sine: \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

These identities are embedded in the verification of our given problem, allowing us to express the sine of a sum or difference in terms of simpler functions.

Sine Function

The sine function is one of the foundational trigonometric functions in mathematics. It relates a given angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Not only is it a key function in trigonometry, but it also appears in various contexts across mathematics and science.

The sine function, often abbreviated as "sin," has unique properties:

• It is periodic with a period of \(2\pi\).
• Its maximum value is 1, and its minimum value is -1.
• The function is odd, meaning \( \sin(-x) = -\sin(x) \).

In our exercise, the sine function is used with sum and difference identities to simplify and verify the expression. By expressing more complex angles in terms of simpler sums and differences, we use the properties of the sine function to transform and work with trigonometric identities more easily.

Tangent Function

The tangent function, another critical trigonometric function, is defined as the ratio of the sine function to the cosine function. In mathematical terms, it can be expressed as \( \tan x = \frac{\sin x}{\cos x} \). This definition is crucial to understanding and transforming trigonometric expressions involving tangent.

The properties of the tangent function include:

• Periodicity with a period of \(\pi\).
• Being undefined where the cosine of the angle is zero, causing vertical asymptotes in its graph.
• It is an odd function, which means \( \tan(-x) = -\tan(x) \).

In the context of the given problem, the tangent function is intertwined with the sine function to verify the identity. By substituting the tangent definition into the sum and difference equations, we simplify the expression, enabling a clearer path towards verifying the identity.

Verify Identity

Verifying a trigonometric identity is the process of showing that both sides of the equation are equivalent, using algebraic manipulations and known identities. It's like solving a puzzle, where each piece is moved, transformed, or simplified until both sides match perfectly.

To verify the identity given in our problem, we follow several steps:

• Apply the sum and difference identities for sine as a starting point.
• Substitute the definition of the tangent function, transforming sections of the equation.
• Simplify both sides using arithmetic and algebraic techniques.

Once these steps are performed and the equation becomes balanced, the identity is verified. Verification not only checks correctness but also deepens understanding of how trigonometric functions interact within complex expressions.