Problem 39 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \sin (\alpha+\beta)+\sin (\alpha-\beta)=2 \sin \alpha \cos \beta $$

Step-by-Step Solution

Answer

The identity \(\sin (\alpha+\beta)+\sin (\alpha-\beta)=2 \sin \alpha \cos \beta\) is verified by substituting sum and difference identities for sine, simplifying the expression, and comparing the resulting expression to the right hand side of the identity.

Step 1: Write the left side in terms of sum and difference identities

The sum and difference identities for sine 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 left side of the identity, we get \((\sin \alpha \cos \beta + \cos \alpha \sin \beta) + (\sin \alpha \cos \beta - \cos \alpha \sin \beta)\)

Step 2: Simplify the expression

The terms \(\cos \alpha \sin \beta\) in the expansion cancel out, because they appear once with a positive sign and once with a negative sign. Hence, the expression simplifies to \(2 \sin \alpha \cos \beta\)

Step 3: Compare the simplified expression with the right side

Our simplified expression from Step 2, \(2 \sin \alpha \cos \beta\), matches exactly the right side of the given identity, so the identity has been verified

Key Concepts

Sine Identity

When it comes to understanding trigonometry, being familiar with fundamental identities is key. Let's take a closer look at the sine identity and how it relates to the sum and difference of angles. The sine of a sum, for instance, \(\sin(\alpha + \beta)\), can be expressed using an identity as \(\sin \alpha \cos \beta + \cos \alpha \sin \beta\). Similarly, for the sine of a difference \(\sin(\alpha - \beta)\), the identity is \(\sin \alpha \cos \beta - \cos \alpha \sin \beta\). These expressions are invaluable as they allow us to break down more complex trigonometric problems into manageable pieces.

Recognizing and applying these basic sine identities helps demystify larger trigonometric expressions and is the cornerstone of solving many trigonometric equations. Having a solid grasp of these identities not only aids in solving textbook exercises but also fortifies an understanding of sine's behavior with respect to angle changes.

Trigonometric Identities

Trigonometric identities are essentially math's version of a Swiss Army knife for trigonometry. They are equations that hold true for all values of the involved variables. Beyond the sine identities mentioned above, there are numerous other trigonometric identities, including those for cosine, tangent, cotangent, secant, and cosecant, as well as Pythagorean identities, double angle identities, and more.

These identities are a set of tools in your mathematical toolbox that you can use to simplify, transform, and solve trigonometric expressions and equations. Understanding how and when to implement these can significantly streamline solving complex problems. For instance, knowing that \(\sin^2\alpha + \cos^2\alpha = 1\) or how to express \(\tan\alpha\) in terms of sine and cosine can make huge differences in solving exercises efficiently.

Identity Verification

Verifying an identity means showing that two different looking expressions are indeed the same for all values within their domains. This often involves transforming one or both sides of an equation until they match. It’s a bit like working with a puzzle where you re-arrange pieces until you find a clear picture, here the clear picture being the simplified form that matches both sides.

To verify an identity like \(\sin(\alpha + \beta) + \sin(\alpha - \beta) = 2\sin\alpha\cos\beta\), we utilize known identities and algebraic manipulation so that eventually, through a series of steps, we witness the initial complex expression transforming and matching the simpler one. Identity verification not only tests one's understanding of trigonometric identities but also hones algebraic skills.

Algebraic Simplification

Algebraic simplification is a fundamental process in mathematics where we aim to reduce expressions to their simplest form. The steps involved may include expanding expressions, combining like terms, factoring, and canceling out terms. In trigonometry, this often employs adding or subtracting terms, and rationalizing denominators, among other techniques.

Simplification is not just about making an expression look 'neater'; it is about making it more understandable and functional for solving problems. For instance, when we simplify \(\sin\alpha\cos\beta + \cos\alpha\sin\beta) + (\sin\alpha\cos\beta - \cos\alpha\sin\beta)\), the cancellations lead us to the elegant expression \(2\sin\alpha\cos\beta\), highlighting algebraic simplification's key role in identity verification.