Problem 46 - Algebra 2

Question

Rewrite each expression as a trigonometric function of a single angle measure. $$ \cos 3 \theta \cos 4 \theta-\sin 3 \theta \sin 4 \theta $$

Step-by-Step Solution

Answer

\(\cos 7 \theta\)

Step 1: Identify the Trigonometric Identity

This expression seems complicated initially. Reviewing trigonometric identities, one can realize that the given expression is actually in the form of the cosine of the sum of two angles, \(a + b\), where the formula is defined as \(\cos (a + b) = \cos a \cos b - \sin a \sin b\). Here, \(a = 3 \theta\) and \(b = 4 \theta\).

Step 2: Apply the Identity

Substitute \(3 \theta\) for \(a\) and \(4 \theta\) for \(b\) in the cosine formula of the sum of two angles. It comes out as \(\cos (3 \theta + 4 \theta)\).

Step 3: Simplify the Expression

Add the two angles together to simplify the expression. The sum is \(7 \theta\). Therefore the expression simplifies to \(\cos 7 \theta\).

Key Concepts

Cosine of Sum of Angles

The concept of adding angles in trigonometry allows us to use trigonometric identities that help simplify expressions. One such identity is the **cosine of the sum of angles**. This concept is primarily used to convert an expression involving multiple trigonometric terms into one involving a single term. In this context, the cosine of sum identity is expressed as:

• \[\cos(a + b) = \cos a \cos b - \sin a \sin b\]

When you have an expression like \( \cos 3\theta \cos 4\theta - \sin 3\theta \sin 4\theta \), it fits perfectly into this identity. By identifying \( a = 3\theta \) and \( b = 4\theta \), we immediately see this expression is a re-arrangement of the cosine of sum formula. Applying this, we can quickly simplify the entire expression to a single term: \( \cos(3\theta + 4\theta) \). This is not only a simplification but also provides a very powerful tool for computations in trigonometry.

Single Angle Trigonometric Expression

In trigonometry, expressions involving multiple trigonometric terms can often be simplified into a single angle trigonometric expression. This is particularly useful when dealing with complex formulas, enabling easier computation and interpretation of trigonometric problems. For our specific problem, through the use of the cosine of sum identity, we've turned an expression with initial angles \( 3\theta \) and \( 4\theta \) into a single angle, represented as \( \cos 7\theta \). The benefit here is clear:

• Simplified computation
• Easier to interpret or graph the trigonometric function

Thus, turning complex trigonometric expressions into a single angle function not only simplifies the math but also improves clarity in understanding the problem as a whole. This conversion is a common and effective technique in solving many different types of trigonometry problems.

Trigonometric Simplification

Trigonometric simplification involves using identities and formulas to reduce complex trigonometric expressions into simpler forms. It is an essential skill in mathematics, facilitating the solving of trigonometric equations and the analysis of mathematical models. In the provided exercise, simplification was accomplished through:

• Identifying the trigonometric identity that the expression fits into. In this case, the cosine of sum of angles.
• Substituting the given angles into the identity formula, transforming it into a single expression \( \cos 7\theta \).
• Combining the angles and simplifying the terms.

Effective use of trigonometric simplification can make solving problems quicker and more intuitive, turning potentially cumbersome expressions into manageable ones. This approach not only saves time but also strengthens your understanding of trigonometric properties and their applications.