Problem 68 - Precalculus Mathematics for Calculus

Question

Find the value of the product or sum. $$3 \cos 37.5^{\circ} \cos 7.5^{\circ}$$

Step-by-Step Solution

Answer

The value is \(\frac{3}{4}(\sqrt{2} + \sqrt{3})\).

Step 1: Recognize the Trigonometric Product to Sum Formula

The product-to-sum identities in trigonometry allow us to rewrite products of sines and cosines into sums. The relevant identity for this problem is:\[ \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right) \] This will help us simplify the product \(3 \cos 37.5^{\circ} \cos 7.5^{\circ}\).

Step 2: Apply the Product to Sum Formula

Using the identity from Step 1, replace \(A\) with \(37.5^{\circ}\) and \(B\) with \(7.5^{\circ}\):\[ \cos 37.5^{\circ} \cos 7.5^{\circ} = \frac{1}{2} \left( \cos(37.5^{\circ} + 7.5^{\circ}) + \cos(37.5^{\circ} - 7.5^{\circ}) \right) \]

Step 3: Calculate the Angles in the Formula

Compute the angles within the cosine functions:\(37.5^{\circ} + 7.5^{\circ} = 45^{\circ}\)\(37.5^{\circ} - 7.5^{\circ} = 30^{\circ}\)

Step 4: Simplify using Known Cosine Values

Substitute the cosine values into the expression:\[ \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 30^{\circ} = \frac{\sqrt{3}}{2} \]The expression becomes:\[ \cos 37.5^{\circ} \cos 7.5^{\circ} = \frac{1}{2} \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \right) \]

Step 5: Combine and Simplify the Expression

Simplify the expression:\[ \frac{1}{2} \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \right) = \frac{1}{4}(\sqrt{2} + \sqrt{3}) \]

Step 6: Multiply by the Coefficient

Multiply the simplified expression by 3:\[ 3 \times \frac{1}{4}(\sqrt{2} + \sqrt{3}) = \frac{3}{4}(\sqrt{2} + \sqrt{3}) \]

Step 7: Final Answer

Therefore, the value of the given product is:\[ \frac{3}{4}(\sqrt{2} + \sqrt{3}) \]

Key Concepts

Product-to-Sum Identities

Trigonometry often involves complex expressions that can be simplified using identities. One such powerful tool is the product-to-sum identities. These identities transform products of trigonometric functions into sums, which are often easier to calculate.
This can be particularly helpful when you're working with angles that are not common on trigonometric tables.For instance, the identity for the product of two cosines is:

• \( \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right) \)

By using this identity, you can transform a product involving two angles into an expression that involves only additions or subtractions within cosine functions.
This simplified form often employs angles for which cosine values are well known or can be easily computed.

Cosine Values

Cosine values are a foundational aspect of trigonometry. When you solve problems involving trigonometric functions, knowing standard cosine values is tremendously beneficial.
These standard angles and their cosine values appear frequently across many types of problems, including those using product-to-sum identities.Some of the most important cosine values include:

• \( \cos 0^{\circ} = 1 \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \cos 90^{\circ} = 0 \)

Utilizing these values helps streamline the process of simplifying expressions.
In many cases, as angles are transformed using identities, the new angles are ones for which these standard values apply, facilitating easier computations.

Angle Calculation

Angle calculations in trigonometry can sometimes seem complicated, but they are often straightforward when using trigonometric identities.
This process involves basic arithmetic operations to find new angles that are crucial in simplifying expressions.In the exercise, we started with angles:\(37.5^{\circ}\) and \(7.5^{\circ}\).
By using the product-to-sum formula:

• The sum of the angles: \(37.5^{\circ} + 7.5^{\circ} = 45^{\circ}\)
• The difference of the angles: \(37.5^{\circ} - 7.5^{\circ} = 30^{\circ}\)

Both \(45^{\circ}\) and \(30^{\circ}\) are standard angles for which cosine values are well-known.
This makes them ideal for simplifying trigonometric expressions, as you can easily substitute these using cosine tables or memory.