Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the product-to-sum formulas to write the product as a sum or difference. $$3 \sin 2 \alpha \sin 3 \alpha$$

Step-by-Step Solution

Answer

The expression \(3 \sin 2 \alpha \sin 3 \alpha\) can be written as the difference of cosines \((3/2) [\cos \alpha - \cos(5 \alpha)]\).

Step 1: Recall the Product-to-Sum identities

Two of the product-to-sum identities are \(2 \sin a \sin b = \cos(a - b) - \cos(a + b)\) and \(2 \cos a \cos b = \cos(a - b) + \cos(a + b)\). Here, we notice the coefficients of \(\alpha\) are different; therefore, the first identity should be used.

Step 2: Rewrite the expression with a coefficient of 2

We should first rewrite the original expression properly so it can align with the product-to-sum identity formula. So, we rewrite the expression as \(2(3/2) \sin 2 \alpha \sin 3 \alpha\). Here, we just factored out 2 from 3. This will work for the first product-to-sum identity.

Step 3: Apply Product-to-Sum formula

Now, replace \(a\) with \(2 \alpha\) and \(b\) with \(3 \alpha\) in the product-to-sum formula. Applying the formula yields \((3/2) [\cos(2 \alpha - 3 \alpha) - \cos(2 \alpha + 3 \alpha)]\)

Step 4: Simplify the expression

Simplify the expression to get \((3/2) [\cos(- \alpha) - \cos(5 \alpha)]\). Due to the property that cosine is an even function (\(\cos(- \theta) = \cos \theta\)), we can further rewrite the expression as \((3/2) [\cos \alpha - \cos(5 \alpha)]\).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are a very useful tool in trigonometry. They're used to convert the product of trigonometric functions into a sum or difference. This can simplify complex expressions and make solving trigonometric equations easier.

Here's the key formula for the sine function:

• \(2 \sin a \sin b = \cos(a - b) - \cos(a + b)\)
• \(2 \cos a \cos b = \cos(a - b) + \cos(a + b)\)

In the exercise you worked on, the formula \(2 \sin a \sin b = \cos(a - b) - \cos(a + b)\) was used. This formula changes the product of two sine functions into a difference of two cosine functions.

This is very helpful because it can transform the problem into something more manageable. By changing products into sums or differences, complicated trigonometric identities become simpler. This is not only handy for solving equations but also for integrating trigonometric expressions.

Trigonometric Functions

Trigonometric functions form the backbone of trigonometry. The basic ones are sine, cosine, and tangent. They arise in the context of right-angled triangles, where they relate the angles to the ratios of the triangle's sides. Here’s a quick reminder of each:

• Sine (sin): Opposite side over the hypotenuse.
• Cosine (cos): Adjacent side over the hypotenuse.
• Tangent (tan): Opposite side over the adjacent side.

These functions are not only useful in geometry, but also appear in waves, circles, and oscillating phenomena.

In the context of the exercise, sine and cosine are used. We multiply two sine functions together, and then use a product-to-sum formula to convert this product into a sum of cosine functions. Understanding how different trigonometric functions relate to each other is crucial for mastering more advanced topics.

Cosine Function

The cosine function, often written as \(\cos(\theta)\), is one of the fundamental functions in trigonometry. Cosine represents the ratio of the length of the adjacent side to the hypotenuse in a right-angle triangle. It is known for several key properties:

• Cosine is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• It has a range of [-1, 1] and is periodic with a period of \(2\pi\).
• Cosine values repeat their pattern every \(360^\circ\) or \(2\pi\) radians.

The fact that cosine is an even function was used in simplifying the expression in your exercise. Initially, we had \(\cos(-\alpha)\), and using the even property, it simplified to \(\cos(\alpha)\).

Being familiar with these properties can significantly aid in problem-solving, helping you transform and simplify trigonometric expressions with ease.