Problem 160 - Precalculus

Question

For the following exercises, prove the identities. $$ \sin (16 x)=16 \sin x \cos x \cos (2 x) \cos (4 x) \cos (8 x) $$

Step-by-Step Solution

Answer

The identity is proven by expressing \( \sin(16x) \) using successive double angle formulas.

Step 1: Understanding the Problem

We are given the trigonometric identity \( \sin(16x) = 16 \sin x \cos x \cos(2x) \cos(4x) \cos(8x) \) and need to prove that both sides of this equation are equal for any angle \( x \). This involves showing how the left side, \( \sin(16x) \), can be transformed into the right side, or vice versa.

Step 2: Exploring Left Side

Start with the left side of the equation, \( \sin(16x) \). We need to express \( \sin(16x) \) using product-to-sum or double-angle identities. Notice that \( 16x \) suggests repetitive use of the double angle formula for sine: \( \sin(2\theta) = 2\sin\theta\cos\theta \).

Step 3: Recursive Double Angle Formula Application

Apply the double angle formula repeatedly:- \( \sin(16x) = 2\sin(8x)\cos(8x) \)- \( \sin(8x) = 2\sin(4x)\cos(4x) \)- \( \sin(4x) = 2\sin(2x)\cos(2x) \)- \( \sin(2x) = 2\sin(x)\cos(x) \)These expressions accumulate progressively.

Step 4: Synthesizing Previous Results

Substitute these identities progressively:\( \sin(16x) = 2 \times 2 \times 2 \times 2 \times \sin x \times \cos x \times \cos(2x) \times \cos(4x) \times \cos(8x) \).This results in \( 16 \sin x \cos x \cos(2x) \cos(4x) \cos(8x) \).

Step 5: Conclusion of Proof

The process shows that \( \sin(16x) \) simplifies exactly to \( 16 \sin x \cos x \cos(2x) \cos(4x) \cos(8x) \). This proves that the initial identity \( \sin(16x) = 16 \sin x \cos x \cos(2x) \cos(4x) \cos(8x) \) holds true.

Key Concepts

Double Angle Formula

Trigonometric functions often require us to work with angles that are multiples of each other. This is where the Double Angle Formula comes into play. It provides a way to simplify expressions that involve trigonometric functions of double angles into manageable terms. Specifically, the double angle formula for sine is \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). This formula allows us to break down expressions like \( \sin(16x) \) into smaller components by continuously applying the formula.

• Start with \( \sin(16x) \). Using the formula, this becomes \( 2\sin(8x)\cos(8x) \).
• Repeating for \( \sin(8x) \), it turns into \( 2\sin(4x)\cos(4x) \).
• Continuing further for \( \sin(4x) \), apply it again, resulting in \( 2\sin(2x)\cos(2x) \).
• Finally, for \( \sin(2x) \), it becomes \( 2\sin(x)\cos(x) \).

This recursive application helps transform \( \sin(16x) \) into a product of angles, making it easier to prove identities or solve equations involving multiple angles.

Product-to-Sum Formulas

The product-to-sum formulas are powerful tools in trigonometry that allow us to convert products of sine and cosine into sums or differences. However, in this particular exercise, we've primarily focused on the double angle identity instead of the product-to-sum directly. Yet, understanding this concept is crucial, especially when faced with expressions involving multiple trigonometric products. When tackling problems, product-to-sum formulas can transform:

• \( \sin A \sin B \) into \( \frac{1}{2} [\cos(A-B) - \cos(A+B)] \).
• \( \cos A \cos B \) into \( \frac{1}{2} [\cos(A+B) + \cos(A-B)] \).
• \( \sin A \cos B \) into \( \frac{1}{2} [\sin(A+B) + \sin(A-B)] \).

In problems like ours, when looking at products, knowing these conversions help check if there's a simpler form revealing insights needed for a proof or solution. It eases the complexity by translating multiplicative elements of a trigonometric identity into more approachable additive and subtractive forms.

Trigonometric Proofs

Trigonometric proofs involve showing that two complex expressions are equivalent using known identities. This process often requires breaking down expressions into known identities and then simplifying both sides step by step. In the given exercise, the goal was to prove \( \sin(16x) = 16 \sin x \cos x \cos(2x) \cos(4x) \cos(8x) \).The strategy includes:

• Identifying and applying relevant identities such as the double angle formula.
• Simplifying one side of the equation into the form of the other side, recognizing patterns such as recurring factors or canceling out terms.
• Checking each step to ensure transformations follow logically from known identities.

By transforming \( \sin(16x) \) and breaking it down through repeated applications of known formulas, the proof becomes clearer and confirms the integrity of the identity provided. This systematic approach is crucial for solving and verifying identities in trigonometry.