Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ \cos x+\cos 2 x $$

Step-by-Step Solution

Answer

\( 2 \cos \left( \frac{3x}{2} \right) \cos \left( \frac{x}{2} \right) \)

Step 1: Recall Trigonometric Identity

We can use a trigonometric identity to express the sum of two cosine terms as a product. Specifically, recall that \( \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).

Step 2: Identify Values of A and B

In the expression \( \cos x + \cos 2x \), we identify \( A = x \) and \( B = 2x \).

Step 3: Apply the Cosine Sum Identity

We substitute \( A = x \) and \( B = 2x \) into the identity from Step 1:\[\cos x + \cos 2x = 2 \cos \left( \frac{x + 2x}{2} \right) \cos \left( \frac{x - 2x}{2} \right)\]

Step 4: Simplify Expressions

Simplify the terms in the cosine identity:- \( \frac{x + 2x}{2} = \frac{3x}{2} \)- \( \frac{x - 2x}{2} = \frac{-x}{2} \)Thus, \[ \cos x + \cos 2x = 2 \cos \left( \frac{3x}{2} \right) \cos \left( \frac{-x}{2} \right) \].

Step 5: Simplify Further if Possible

Using the property \( \cos((-x)) = \cos(x) \), the expression becomes:\[ 2 \cos \left( \frac{3x}{2} \right) \cos \left( \frac{x}{2} \right) \]

Step 6: Summarization Step: Write Final Expression

The expression \( \cos x + \cos 2x \) is equivalent to:\[ 2 \cos \left( \frac{3x}{2} \right) \cos \left( \frac{x}{2} \right) \]

Key Concepts

Cosine Sum Identity

When exploring trigonometry, the cosine sum identity is a valuable tool that allows us to simplify expressions involving the sum of two cosine functions. This identity states:

• For any angles \( A \) and \( B \), \[ \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right). \]

This identity simplifies expressions efficiently and is crucial in converting sums into products, which can often be easier to work with in solving equations or proving other identities.

To apply this identity in the expression \( \cos x + \cos 2x \), we identify \( A = x \) and \( B = 2x \). By substituting these into the formula, we can rewrite the original expression as a product of cosines. It's an effective strategy to reduce complexity or when aiming to integrate or differentiate trigonometric expressions easily.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has profound applications ranging from basic geometry to advanced calculus and science. Some key elements of trigonometry include:

• Understanding the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
• Mastering unit circle definitions, which help in understanding these functions' behavior.
• Learning and applying various identities, such as Pythagorean identities, sum-difference formulas, and double-angle formulas.

These components are essential in fields such as physics, engineering, and computer science, where trigonometric concepts solve real-world problems.

In our exercise, recognizing how to apply trigonometric identities like the cosine sum identity is a critical skill in simplifying expressions and solving equations.

Simplification in Trigonometry

Trigonometric simplification involves transforming complex trigonometric expressions into simpler forms. This makes them easier to analyze and use in calculations, such as solving equations or finding limits.Key techniques for simplification include:

• Using identities like the cosine sum identity to transform sums into products or vice versa.
• Applying algebraic techniques, such as factoring, to break down expressions.
• Recognizing angle reductions: utilizing known values (e.g., \( \cos(-x) = \cos(x) \)) to simplify terms further.

In our specific problem, we saw how the expression \( \cos x + \cos 2x \) was simplified by expressing it as a product: \[ 2 \cos \left( \frac{3x}{2} \right) \cos \left( \frac{x}{2} \right). \]
These simplification strategies are essential for students who need to solve trigonometric equations effectively and are commonly used in calculus and analytic geometry.