Problem 17 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \cos \frac{3 x}{2}+\cos \frac{x}{2} $$

Step-by-Step Solution

Answer

So, the expression \(\cos \frac{3 x}{2}+\cos \frac{x}{2}\) can be re-written as \(2 \cos^2 x\).

Step 1: Recognize and apply the trigonometric formula

Express the given sum of cosines as a product using the formula \(2 \cos \frac{{a+b}}{2} \cos \frac{{a-b}}{2} = \cos a + \cos b\). In this problem, \(a = \frac{3x}{2}\) and \(b = \frac{x}{2}\). So, plugging these into the formula, you should get \(2 \cos x \cos x = \cos \frac{3x}{2} + \cos \frac{x}{2}\)

Step 2: Simplify the expression

The expression \(2 \cos x \cos x\) simplifies to \(2 \cos^2 x\).

Key Concepts

Sum-to-Product Formulas

Sum-to-Product formulas are essential tools in trigonometry that help simplify expressions involving trigonometric functions. They allow us to express sums or differences of sines and cosines as products. This transformation can make integration or solving equations more straightforward.

For example, the formula \(\cos a + \cos b = 2 \cos \frac{{a+b}}{2} \cos \frac{{a-b}}{2}\) converts the sum of two cosines into a product of cosines. These formulas are useful because they condense trigonometric expressions, making them easier to manipulate and evaluate.

When using these formulas, always check the specific angles involved. In practice, for our original problem, we have \(a = \frac{3x}{2}\) and \(b = \frac{x}{2}\), leading us to express the sum as \(2 \cos x \cos x\). This approach simplifies the problem significantly, as the next steps focus on algebraic manipulation.

Cosine Function

The cosine function, one of the fundamental trigonometric functions, is often encountered in trigonometry questions. It describes the relationship between the angle of a right triangle and the length of the adjacent side over the hypotenuse.

When dealing with angles in the form of \(\frac{3x}{2}\) and \(\frac{x}{2}\), it's crucial to understand how these affect the cosine values. The cosine function has a periodic nature, with a period of \(2\pi\), allowing values to repeat every full cycle.

The original problem deals with angles expressed in terms of \(x\). Understanding how the cosine function behaves with different expressions of angle is pivotal in applying formulas correctly. By grasping how changes in angle expressions impact the function, such as transforming sum to product, we can predict and simplify outcomes accurately.

Algebraic Simplification

Algebraic simplification is a process used to transform expressions into simpler or more manageable forms. In trigonometry, this process often involves reducing complex expressions after applying trigonometric identities and formulas.

In the given exercise, after applying the sum-to-product formula, we arrive at \(2 \cos x \cos x\). The expression can be simplified further because like terms appear. Since \(\cos x \cos x\) is \(\cos^2 x\), the expression simplifies to \(2 \cos^2 x\).

Understanding algebraic simplification is essential for efficient problem-solving, as it allows us to condense expressions to forms that are easier to interpret or evaluate. This simplification process is commonly followed by checking whether further factorization or reduction is possible, eventually offering a more elegant or insightful mathematical view of the original problem.