Problem 59 - Precalculus Essentials

Question

Verify the given sum-to-product formula. Start with the right side and obtain the expression on the left side by using an appropriate product-to-sum formula. $$\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$$

Step-by-Step Solution

Answer

After applying the product-to-sum formula to expression on the right side of the equation, and simplifying the result, we can confirm that the sum-to-product formula given is correct.

Step 1: Locate Formulas

First step is to recall the product-to-sum formulas, which are: \[\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]\] These formulas will be used to convert the products on the right side of the equation to sums.

Step 2: Apply Formula

Apply the product-to-sum formula to the expression on the right side of the equation: \[\frac{1}{2}[\cos((\alpha+\beta)/2 - (\alpha-\beta)/2) + \cos((\alpha+\beta)/2 + (\alpha-\beta)/2)]\] Keeping in mind that the 2 in the 2 cos expression gets multiplied with the result.

Step 3: Simplify Result

Fix up the expression to simplify the sum and difference of the fractions in the argument the cosine function: \[= \frac{1}{2}[\cos(2\alpha/2) + \cos(2\beta/2)] = \frac{1}{2}[2\cos\alpha + 2\cos\beta] = \cos\alpha + \cos\beta\] This simplification confirms that the original sum-to-product formula is correct.

Key Concepts

Product-to-Sum Formulas

When working with trigonometric functions, there are handy formulas known as product-to-sum formulas that help in transforming products of trigonometric functions into sums or differences.

• These formulas simplify computations and aid in verifying identities, like sum-to-product transformations.
• For cosine functions specifically, the product-to-sum formula is: \[ \cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)] \]

This formula is essential, especially when dealing with trigonometric expressions that may initially appear complex. By applying this formula, you can simplify calculations and make connections easier to identify. In exercises like the original one, we use the product-to-sum formula to convert expressions into a form that's simpler or matches another given format. The step-by-step application of the formula transforms complex product expressions into ones that are easier to manage and verify.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. They are fundamental in simplifying and solving trigonometric problems, and they can transform complex expressions into simpler ones.

• One basic category includes Pythagorean identities like \( \sin^2 A + \cos^2 A = 1 \).
• There are angle sum and difference identities, such as \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \).
• Intricate problems often involve using multiple identities in combination.

For example, the step-by-step solution you reviewed used a specific identity — a product-to-sum formula — that enabled the transformation of the trigonometric expression.Trigonometric identities like these are powerful tools to solve equations, prove other identities, and can simplify functions in calculus. They offer versatility and make seemingly intimidating expressions more manageable.

Cosine Addition Formula

The cosine addition formula is a trigonometric identity that expresses \( \cos(A + B) \) using the cosine and sine of \( A \) and \( B \).

• The formula is given by: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

This identity is crucial whenever you need to break down cosine of a sum into a sum of products, which can be a key step in many proofs and transformations. Though it wasn’t explicitly part of the step-by-step solution for the given exercise, understanding it is beneficial for comprehending how angles combine in various trigonometric operations. Much like the product-to-sum formulas, the cosine addition formula serves to simplify or rearrange expressions, particularly in scenarios involving addition of angles. It’s a valued identity within the broader context of trigonometry, often appearing in applications ranging from geometry to complex number theory.