Problem 69 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified single term expression for \(\cos(\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta\) is \(\cos\alpha\).

Step 1: Recognize the identities present

You can recognize the form of sine and cosine of sum of two angles in the given expression. This is given as: \(\cos(\alpha + \beta)\cos\beta + \sin(\alpha + \beta)\sin\beta\)

Step 2: Apply Cosine of Sum identity

The first term of the expression \(\cos(\alpha + \beta)\cos\beta\) looks like the part of the formula for the cosine of a sum of angles, \(\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta\). Let's replace \(\cos(\alpha + \beta)\) with this identity: \( (\cos\alpha \cos\beta - \sin\alpha \sin\beta) \cos\beta\)

Step 3: Apply Sine of Sum identity

Similarly, the second term of the expression \(\sin(\alpha + \beta)\sin\beta\) resembles the part of formula for the sine of a sum of angles, \(\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\). Replace \(\sin(\alpha + \beta)\) with this identity: \( (\sin\alpha \cos\beta + \cos\alpha \sin\beta) \sin\beta\)

Step 4: Simplify the expression

Add these expressions together and simplify: \( (\cos\alpha \cos\beta - \sin\alpha \sin\beta) \cos\beta + (\sin\alpha \cos\beta + \cos\alpha \sin\beta) \sin\beta\). On simplifying and arranging the terms, obtain \(\cos\alpha(\cos^2\beta + \sin^2\beta)\). We know that \(\cos^2\beta + \sin^2\beta = 1\), so the final expression simplifies to \(\cos\alpha\).

Step 5: Conclusion

So the given expression can be simplified to \(\cos\alpha\). This means that the original expression \(\cos(\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta\) can be represented in one term as \(\cos\alpha\).

Key Concepts

Cosine of Sum Identity

Understanding the cosine of sum identity is a fundamental part of trigonometry that allows us to simplify expressions involving the cosine of a sum of two angles. The identity is expressed as:

\[\begin{equation} \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta \end{equation}\]
This identity implies that the cosine of the sum of two angles can be broken down into a combination of the individual cosines and sines of those angles. When facing a trigonometric problem, identifying structures that resemble this identity is crucial. By substituting \(\cos(\alpha + \beta)\) with its equivalent expression from the identity, we can simplify complex trigonometric expressions into simpler forms.

Sine of Sum Identity

Similarly, the sine of sum identity is instrumental when working with the sine of the addition of two angles. It is given by:

\[\begin{equation} \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \end{equation}\]
This identity shows that the sine of a sum is the sum of the product of the sine and cosine of the individual angles. Recognizing this pattern allows students to transform a given expression into a more manageable form, as was done with the identity for cosine. Substituting \(\sin(\alpha + \beta)\) with its identity during calculations facilitates the process of simplifying trigonometric terms.

Simplifying Trigonometric Expressions

The process of simplifying trigonometric expressions is all about recognizing patterns and applying identities effectively. Key to this concept is understanding the fundamental trigonometric identities like \(\sin^2\beta + \cos^2\beta = 1\), which states that the square of the sine of an angle plus the square of the cosine of the same angle always equals one. Utilizing this core identity can turn seemingly complicated expressions into much simpler ones. During simplification, it is essential to combine like terms, factor common factors, and cancel out terms where appropriate to achieve the most reduced form of the expression. This step-by-step methodical approach transforms a complex expression into a basic function of a single angle or variable.

Angle Addition Formulas

The angle addition formulas encapsulate both the sine and cosine of sum identities. These formulas are incredibly powerful as they enable the evaluation of the trigonometric functions for the sum or difference of two angles. Recognizing when to apply these formulas is critical when solving trigonometry problems. The angle addition formulas are not only vital in simplification but also in expanding, evaluating, and even solving more complex trigonometric equations. Having a good grasp of these formulas shades light on the periodic and cyclical nature of trigonometric functions, which is invaluable when dealing with periodic phenomena in various fields of study and real-world applications.