Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a sum or difference. $$ 2 \sin 7 \theta \sin 5 \theta $$

Step-by-Step Solution

Answer

\( \cos(2\theta) - \cos(12\theta) \)

Step 1: Identify the Formula Needed

To express the product of two sine functions as a sum or difference, we can use the product-to-sum identities. The relevant identity for this problem is: \[ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \]

Step 2: Assign Values to A and B

In the given expression, \( A = 7\theta \) and \( B = 5\theta \). These will be used in the product-to-sum identity to transform the product into a sum or difference.

Step 3: Apply the Product-to-Sum Identity

Substitute \( A = 7\theta \) and \( B = 5\theta \) into the identity: \[ 2 \sin(7\theta) \sin(5\theta) = \cos((7\theta) - (5\theta)) - \cos((7\theta) + (5\theta)) \] Simplifying inside the cosine functions gives: \[ \cos(2\theta) - \cos(12\theta) \]

Step 4: Simplify the Expression

The expression is now simplified to \[ \cos(2\theta) - \cos(12\theta) \]. This is the sum or difference expression of the original product of sines.

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are a valuable tool in trigonometry, ideal for transforming the product of trigonometric functions into a sum or difference. These formulas help simplify expressions, making them easier to integrate or differentiate in calculus, and are crucial in signal processing when analyzing waves. In the context of our exercise, we're converting the product of two sine functions into a sum or difference using:
\[2 \sin A \sin B = \cos(A-B) - \cos(A+B)\]
This formula specifically applies to two sine functions. It's derived from the sum and difference formulas for cosine and leverages the periodic nature of trigonometric functions. By using this formula effectively, you can unravel complex expressions and find solutions that are otherwise hard to spot.
For example, transforming \(2 \sin 7\theta \sin 5\theta\) into a sum or difference simplifies to \(\cos 2\theta - \cos 12\theta\), offering a clearer view of the underlying trigonometric relationships.

Sine and Cosine Functions

The sine and cosine functions are fundamental components of trigonometry. Understanding them is key to grasping a wide range of mathematical concepts and real-world applications.

• The sine function, denoted as \(\sin(\theta)\), measures the ratio of the opposite side to the hypotenuse in a right triangle, or simply represents the y-coordinate of a point on the unit circle.
• Cosine, denoted \(\cos(\theta)\), measures the ratio of the adjacent side to the hypotenuse, or the x-coordinate of a point on the unit circle.

These functions also reflect periodic behavior, repeating every 360 degrees (or 2\(\pi\) radians). They are interconnected through various trigonometric identities that provide shortcuts in calculations and insights into the geometry of circles.
In scenarios involving transformations of products to sums, understanding how sine and cosine interact allows manipulation of complex waveforms, aiding in problem-solving strategies or engineering contexts.

Angle Transformations

Angle transformations in trigonometry involve altering the angles within trigonometric functions to simplify expressions or to evaluate them under different conditions. This can include adding, subtracting, or even doubling angles.

• When dealing with angles like \(7\theta\) and \(5\theta\) in trigonometric expressions, correctly transforming these angles is vital for applying identities like the product-to-sum formulas.
• Subtracting angles (as in \(A - B\)) or adding them (\(A + B\)) streamlines the calculation by altering the initial expression into something potentially more manageable.

For instance, in our original transformation, the substitution of \(A = 7\theta\) and \(B = 5\theta\) involves calculating \(A - B = 2\theta\) and \(A + B = 12\theta\), allowing the transformation from a product of sines into a precise difference of cosines. Mastering angle transformations paves the way for solving intricate trigonometric equations and provides a deeper understanding of rotational movement in physics.