Problem 45 - Algebra 2

Question

Rewrite each expression as a trigonometric function of a single angle measure. $$ \sin 3 \theta \cos 2 \theta+\cos 3 \theta \sin 2 \theta $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sin 5 \theta\)

Step 1: Apply the product-to-sum formulas

We apply the addition version of the product-to-sum formula to the given expression. Thus, \[\sin 3 \theta \cos 2 \theta = \frac{1}{2} [\sin(3\theta+2\theta)+\sin(3\theta-2\theta)] = \frac{1}{2} [\sin 5 \theta + \sin \theta]\], and \[\cos 3 \theta \sin 2 \theta = \frac{1}{2} [\sin(3\theta+2\theta)-\sin(3\theta-2\theta)] = \frac{1}{2} [\sin 5 \theta - \sin \theta]\].

Step 2: Combine the expressions

Now, add these two expressions to get:\[ \sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta = \frac{1}{2} [\sin 5 \theta + \sin \theta] + \frac{1}{2} [\sin 5 \theta - \sin \theta]\]

Step 3: Simplify the expression

After adding, the components that are identical on both sides will cancel out, leaving us with: \[\sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta = \sin 5 \theta\]

Key Concepts

Product-to-Sum Formulas

When dealing with trigonometric expressions, the product-to-sum formulas can be exceptionally helpful. These formulas allow you to convert products of sine and cosine functions into sums or differences of trigonometric functions, simplifying complex expressions.

• The basic idea is combining the angles of two trigonometric functions into a single angle, which is useful for further simplification or integration.
• For example, for expressions like \( \sin A \cos B \), you can use the formula: \[ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \]
• This transforms the product of \( \sin A \) and \( \cos B \) into a sum of two sines with different angles.
• Similarly, the formula for \( \cos A \sin B \) is \[ \cos A \sin B = \frac{1}{2} [\sin(A+B) - \sin(A-B)] \]

In the exercise, we specifically applied these to \( \sin 3\theta \cos 2\theta \) and \( \cos 3\theta \sin 2\theta \), converting them into the sum of sines algebraically.

Angle Addition

The angle addition formulas are another crucial tool in trigonometry. They allow the computation of the sine, cosine, or tangent of the sum or difference of two angles. This is essential in the simplification of trigonometric expressions and solving equations.

• The angle addition formula for sine is: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
• For cosine, the formula is: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
• These formulas are invaluable for finding the trigonometric value of more complex angles by breaking it down into known values and simpler angles.

In the given exercise, when we applied the product-to-sum formulas, we essentially used decomposed angle sum formulas to combine \( 3\theta \) and \( 2\theta \), transforming them into a single trigonometric function \( \sin 5\theta \). This illustrates how angle addition and product-to-sum formulas often work hand-in-hand.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are fundamental in understanding triangle properties, wave behaviors, and circular motion. Understanding these functions is key to mastering trigonometry and calculus applications.

• These functions are periodic, meaning they repeat their values in regular intervals, making them ideal for modeling cyclic phenomena such as sound and light waves.
• Important identities involving these functions, such as Pythagorean identities, help simplify and solve many mathematical problems.
• In our exercise, \( \sin 3\theta \), \( \cos 2\theta \), \( \cos 3\theta \), and \( \sin 2\theta \) are combined and simplified using trigonometric identities to reach an expression in terms of a single angle, \( \sin 5\theta \).

Recognizing how to manipulate and transform trigonometric expressions, as done in the exercise, illustrates the power of trigonometric identities and functions in simplifying complex expressions. With practice, these processes become intuitive, aiding in more advanced mathematical problems.