Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ 2 \sin 9 \theta \cos 3 \theta $$

Step-by-Step Solution

Answer

The expression is \( \sin 12\theta + \sin 6\theta \).

Step 1: Identify the formula

The given expression is \( 2 \sin 9\theta \cos 3\theta \). The objective is to express it as a sum or difference. This can be accomplished by using the trigonometric identity for the product-to-sum formula: \( 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \).

Step 2: Apply the formula

Now substitute \( A = 9\theta \) and \( B = 3\theta \) into the identity to transform the expression:\( 2 \sin 9\theta \cos 3\theta = \sin(9\theta + 3\theta) + \sin(9\theta - 3\theta) \).

Step 3: Simplify the expression

Calculate both expressions inside the sine functions:\( \sin(9\theta + 3\theta) = \sin(12\theta) \),and\( \sin(9\theta - 3\theta) = \sin(6\theta) \).

Step 4: Write the final expression

Thus, the original expression \( 2 \sin 9\theta \cos 3\theta \) can be expressed as:\( \sin 12\theta + \sin 6\theta \).

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas are a set of trigonometric identities that help you convert products of sine and cosine into sums or differences. These formulas are incredibly useful for simplifying expressions that involve the multiplication of trigonometric functions. For example, consider the formula: \[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \]This formula lets you rewrite the product \(2 \sin 9\theta \cos 3\theta\) as a sum of sines. Why is this useful? Well, sometimes these forms make it easier to integrate or differentiate trigonometric functions, or simplify complex expressions.

Whenever you see a product of sine and cosine, think of this identity as a quick hack to transform that into something more manageable. Remember, practice makes perfect. Try applying this identity to different problems to reinforce your understanding. You’ll find that they are incredibly handy in both basic and advanced mathematics.

Trigonometric Expressions

Trigonometric expressions are mathematical expressions that involve the trigonometric functions like sine, cosine, and tangent. In many mathematical problems, you'll encounter such expressions needing simplification or transformation. These expressions can appear in angles, series, or more complex equations.

A typical situation involves turning a multiplication of trigonometric functions into a sum or difference, as we observed in the example:\[ 2 \sin 9\theta \cos 3\theta \]Using the product-to-sum formula, we transformed this into a sum, making calculations easier. It's a bit like finding a shortcut in a maze. When working with these expressions, always consider if a known identity can simplify the task. It reduces complexity and often reveals deeper insights or solutions hidden behind the initial form.

• Look for familiar identities like Pythagorean identities or angle sum and difference identities.
• Use identities to alter expressions into a more usable form.
• Stay consistent with angle measurements in degrees or radians throughout your calculations.

Sine and Cosine Functions

The sine and cosine functions are among the cornerstones of trigonometry, often taught early in math courses. They are periodic functions, meaning they repeat their values in regular intervals. This periodic nature makes them invaluable for modeling waves and oscillations.

In trigonometric expressions, sine and cosine often appear as products when dealing with wave interference or signal processing tasks. For example, in the expression \( 2 \sin 9\theta \cos 3\theta \), both sine and cosine describe periodic phenomena that can be combined into simpler wave forms through identities like product-to-sum.Understanding these functions is crucial not just for trigonometry but for calculus and physics as well. Some key points include:

• The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• Both sine and cosine functions have ranges between -1 and 1.

Master these fundamentals alongside their identities, and you'll be well-equipped to handle increasingly complex mathematical challenges.