Problem 13 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \cos 4 x+\cos 2 x $$

Step-by-Step Solution

Answer

The given trigonometric expression is equivalent to \(2 \cos(3x) \cos(x)\)

Step 1: Identifying the identities

The sum-to-product identities can be used to express sums of trigonometric functions as products. The applicable identity here is \(\cos a + \cos b = 2 \cos \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right)\).

Step 2: Apply the identities

Applying the aforementioned identity to the given expression, the equation \(\cos 4x + \cos 2x\) becomes \(2 \cos\left(\frac{4x + 2x}{2}\right) \cos\left(\frac{4x - 2x}{2}\right)\).

Step 3: Simplify the expression

Simplify the fractions inside both cosines. This yields \(2 \cos(3x) \cos(x)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the realm of geometry and waves. They describe the relationships between the angles and sides of right-angled triangles. Two of the primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each of these functions helps to determine various properties of angles and the orientation of an object:

• Sine (\( \sin \)): This function relates the opposite side to the hypotenuse of a right-angled triangle.
• Cosine (\( \cos \)): This function connects the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): This function is the ratio of the opposite side to the adjacent side.

These functions are not limited to triangles; they are widely used to model periodic phenomena such as sound and light waves. Understanding these basic trigonometric functions is crucial before exploring identities like the sum-to-product identities.

Expressing Sums as Products

The sum-to-product identities are a set of trigonometric identities that transform sums of trigonometric functions into products. This transformation is beneficial in simplifying expressions and solving equations. The specific sum-to-product identity used in the exercise is:

• \( \cos a + \cos b = 2 \cos \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right) \).

For instance, when you apply this identity to the expression \( \cos 4x + \cos 2x \), it helps in rewriting the sum as a product, making it easier to evaluate and understand. The steps to apply this method involve first identifying the angles \(a\) and \(b\), then substituting them into the identity, and finally, simplifying the resulting expression. This approach is particularly useful in calculus and signal processing for simplifying complex trigonometric expressions.

Cosine Identities

Cosine identities are essential formulas in trigonometry that break down complex expressions involving cosine functions into simpler, often more usable forms. The most relevant identity in the exercise is the sum-to-product identity:

• \( \cos a + \cos b = 2 \cos \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right) \).

Cosine identities like these allow mathematicians and engineers to tackle problems involving wave interference and harmonics by transforming potentially cumbersome sums into manageable products. Additionally, cosine identities assist in solving equations where direct approaches may not be feasible. Familiarity with these identities can significantly enhance one’s ability to manipulate and understand trigonometric expressions efficiently.