Problem 70 - Algebra and Trigonometry

Question

Rewrite each expression as a simplified expression containing one term. $$ \sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta $$

Step-by-Step Solution

Answer

The given expression simplifies to \( \sin \alpha \).

Step 1: Identity Expansion

The expression \( \sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta \) is given, which can be recognised as a combined sine and cosine difference identities. Due to the known identity \( \sin(\alpha - \beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta \) adjust this expression to match the formula.

Step 2: Applying Trigonometric Identities

Using 'cosine of difference' and 'sine of difference' identities. Those are: 1. \( \cos(\alpha - \beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta \)2. \( \sin(\alpha - \beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta \)Rearrange these identities to match the formula for \( \sin(\alpha - \beta) \) and \( \cos(\alpha - \beta) \)

Step 3: Simplification

After applying the identities you will notice that the equation simplifies to \( \sin \alpha \). So it proves that \( \sin (\alpha-\beta) \cos \beta+\cos (\alpha-\beta) \sin \beta \) equals to \( \sin \alpha \).

Key Concepts

Sine and Cosine Difference Identities

The sine and cosine difference identities are essential formulas in trigonometry that allow us to simplify expressions and solve trigonometric equations. Understanding these identities can help clarify and convert complex trigonometric expressions into simpler forms.

For the sine difference identity, we have:

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

And for the cosine difference identity:

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

These identities are born from the trigonometric addition formulas but focused on the subtraction of angles. By memorizing these identities, we can identify and efficiently handle expressions like those in our problem, where terms appear in combinations resulting in a compact and recognizable form.

Once you recognize the structure resembling these identities, the simplification becomes a systematic task rather than a mystery, making it crucial to practice their application.

Trigonometric Expression Simplification

Simplifying trigonometric expressions often involves using known identities to reduce complex forms into simpler ones. In our example, we were provided with the expression \( \sin (\alpha-\beta) \cos \beta + \cos (\alpha-\beta) \sin \beta \).

The goal of simplification is to recognize these as parts of a fundamental identity that results in a much simpler and often more intuitive expression. This expression is essentially the expanded version of the identity for \( \sin(\alpha) \):

• The given expression matches \( \sin(\alpha) \) when simplified using the identities \( \sin(\alpha - \beta) \) and \( \cos(\alpha - \beta) \).

Simplifying involves several steps:

• Identify the relevant trigonometric identities that apply to the expression.
• Rearrange or expand according to these identities so terms can be cancelled or combined.
• After identifying, it can just lead directly to a known simpler result, such as a single trigonometric function.

This approach not only simplifies but clarifies the trigonometric relationships at play, allowing for a deeper understanding of how angles and sides of a triangle relate in trigonometric forms.

Combination of Sine and Cosine Functions

Combining sine and cosine functions in expressions is a common aspect of trigonometric simplification. This combination often arises, as seen in our example, where multiple trigonometric identities are at work.

In our case, the expression \( \sin (\alpha-\beta) \cos \beta + \cos (\alpha-\beta) \sin \beta \) brings together sine and cosine functions structured in a way that recalls a known identity. This expression directly maps onto the identity \( \sin(\alpha) \) after correctly applying the differences of sine and cosine.

Understanding how sine and cosine can work together in expressions lets us see beyond the immediate complexity:

• Sine and cosine can complement each other, often leading to simple identities due to their fundamental relations.
• Their combination allows one to condense expressions accurately, which can make complex problems much more approachable.

Recognizing these patterns can speed up problem-solving and provide a neat and concise solution. This harmonious relationship of sine and cosine is a powerful tool in reducing expression complexity and is frequently utilized across many areas of mathematics beyond trigonometry. Studying these combinations improves both your intuitive understanding and your practical skills in addressing trigonometric problems.