Problem 9 - Precalculus: Functions and Graphs

Question

Express as a product. $$\sin 6 \theta+\sin 2 \theta$$

Step-by-Step Solution

Answer

The expression is \( 2\sin(4\theta)\cos(2\theta) \).

Step 1: Recognize the Trigonometric Identity

We aim to express the sum \( \sin 6\theta + \sin 2\theta \) as a product. To do this, we can utilize the sum-to-product identities for sine. The general format of this identity is:\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

Step 2: Identify Values for A and B

In the identity \( \sin A + \sin B \), assign: - \( A = 6\theta \) - \( B = 2\theta \)These values are directly from the problem statement.

Step 3: Calculate \(\frac{A + B}{2}\) and \(\frac{A - B}{2}\)

Compute the expressions: - \( \frac{A + B}{2} = \frac{6\theta + 2\theta}{2} = \frac{8\theta}{2} = 4\theta \) - \( \frac{A - B}{2} = \frac{6\theta - 2\theta}{2} = \frac{4\theta}{2} = 2\theta \)

Step 4: Substitute into the Identity

Replace \( \frac{A + B}{2} \) and \( \frac{A - B}{2} \) into the sum-to-product identity:\[ \sin 6\theta + \sin 2\theta = 2 \sin(4\theta) \cos(2\theta) \]

Step 5: Final Solution Verification

Verify that the solution satisfies the original problem statement and is consistent with trigonometric identities. The original sum \( \sin 6\theta + \sin 2\theta \) is now expressed as a product in the form \( 2\sin(4\theta)\cos(2\theta) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with angles and periodic phenomena. They involve relationships between the sine, cosine, tangent, and other trigonometric functions. These identities simplify expressions and solve equations involving trigonometric functions. For instance, in the given exercise, we utilize one called the sum-to-product identity. These identities originate from the addition and subtraction formulas of trigonometric functions and help to convert sums into products or vice versa.
Some of the most popular trigonometric identities include:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Angle Sum and Difference Identities: such as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• Double Angle Formulas: \( \sin 2\theta = 2\sin\theta\cos\theta \)

These identities can be applied whenever you recognize similar structural patterns in trigonometric expressions. Understanding and recognizing these patterns allow you to transform complex trigonometric expressions into simpler forms.

Sine Function

The sine function is a fundamental concept in trigonometry. It represents the y-coordinate or vertical component of a point on the unit circle as the angle varies. The sine function is defined for all real numbers, and its output lies within the interval \([-1, 1]\). This range represents how high or low the y-coordinate can get on the unit circle.
The sine of an angle \(\theta\) in a right-angled triangle can be defined as the ratio of the length of the opposite side to the hypotenuse. In different contexts, such as the unit circle and waveform analysis, the sine function maintains its significance:

• Periodicity: The sine function is periodic with a period of \(2\pi\), meaning \(\sin(\theta + 2\pi) = \sin\theta\).
• Symmetry: It is an odd function, demonstrating symmetry about the origin: \(\sin(-\theta) = -\sin\theta\).

These properties make it a versatile function applicable to both pure and applied mathematics, including transformations and wave analysis.

Product-to-Sum Identities

Product-to-sum identities are another set of trigonometric formulas that complement the sum-to-product identities. They are crucial when converting the product of trigonometric functions into a sum or difference. These identities are derived from the angle addition and subtraction identities, and they help transform functions into more manageable forms.

• Basic Format: For example, \[ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \].
• Key Usage: These identities are especially useful in calculus and physics when integrating trigonometric functions or solving differential equations.

In the exercise, we used the sum-to-product version of these identities to simplify a trigonometric sum into a product. Both product-to-sum and sum-to-product identities enhance the toolbox one can use to manipulate trigonometric expressions, particularly beneficial in scenarios involving complex oscillatory behaviors in physics or engineering.