Problem 82 - Algebra and Trigonometry

Question

Use the identity $$\sin 2 x=2 \sin x \cos x$$ \(n\) times to show that $$\sin \left(2^{n} x\right)=2^{n} \sin x \cos x \cos 2 x \cos 4 x \cdots \cos 2^{n-1} x$$

Step-by-Step Solution

Answer

The identity expands to \(\sin(2^n x) = 2^n \sin x \prod_{j=0}^{n-1} \cos(2^j x)\) by repeated application.

Step 1: Understanding the Identity

We start with the given identity \(\sin 2x = 2\sin x \cos x\). This identity expresses \(\sin 2x\) in terms of \(\sin x\) and \(\cos x\). It can be seen as a basic building block that we will use repeatedly.

Step 2: Applying the Identity Once

First, apply the identity to find an expression for \(\sin(4x)\). Using the identity, we have \(\sin(4x) = 2\sin(2x)\cos(2x)\). Substitute the expression for \(\sin(2x) = 2\sin x \cos x\) into this equation: \(\sin(4x) = 2(2\sin x \cos x)\cos(2x) = 4\sin x \cos x \cos(2x)\).

Step 3: Applying the Identity for General n

Assume by induction that \(\sin(2^k x) = 2^k \sin x \prod_{j=0}^{k-1} \cos(2^j x)\). Now apply the identity to \(\sin(2^{k+1} x) = 2\sin(2^k x)\cos(2^k x)\). Using the induction hypothesis, substitute \(\sin(2^k x)\) with \(2^k \sin x \prod_{j=0}^{k-1} \cos(2^j x)\) to get \(\sin(2^{k+1} x) = 2(2^k \sin x \prod_{j=0}^{k-1} \cos(2^j x))\cos(2^k x)\). This simplifies to \(2^{k+1} \sin x \prod_{j=0}^{k} \cos(2^j x)\).

Step 4: Conclusion for General n

By mathematical induction, we can conclude that the pattern holds for all natural numbers \(n\): \(\sin(2^n x) = 2^n \sin x \prod_{j=0}^{n-1} \cos(2^j x)\). Thus, every time \(\sin\) doubles its angle, we add a \(\cos\) term to the product, and the power of 2 increases by one.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is crucial in understanding various mathematical concepts. It relates to the ratio of the length of the opposite side over the hypotenuse in a right triangle. The sine function is typically denoted as \( \sin(\theta) \), where \( \theta \) is the angle. In a unit circle, the sine value represents the y-coordinate of the point where the terminal side of the angle intersects the circle.

Key properties of the sine function include:

• Its range from -1 to 1.
• It is an odd function, meaning \( \sin(-x) = -\sin(x) \).
• It is periodic with a period of \( 2\pi \).

These characteristics make the sine function invaluable in modeling periodic phenomena like waves and oscillations. Understanding the sine function is an essential step towards mastering trigonometric identities and their applications.

Cosine Function

The cosine function is another primary function in trigonometry, typically paired with the sine function to form the essentials of circular functions. It is noted as \( \cos(\theta) \), and it describes the ratio of the adjacent side over the hypotenuse in a right triangle. For a unit circle, the cosine of an angle represents the x-coordinate of the intersection point with the terminal side of the angle.

Important features of the cosine function include:

• A range from -1 to 1.
• Being an even function, expressed as \( \cos(-x) = \cos(x) \).
• Periodic with a period of \( 2\pi \).

These traits make the cosine function critical for solving problems involving cyclic patterns and for working through trigonometric identities, just like the sine function. The intimacy between sine and cosine functions is shown by the identity \( \sin^2(x) + \cos^2(x) = 1 \), demonstrating how these functions richly complement each other.

Induction Proof

Induction is a mathematical proof technique used to prove statements for all natural numbers. An induction proof normally involves two main steps: a base case and an inductive step. The base case verifies the statement for an initial value, typically for \( n = 0 \) or \( n = 1 \). The inductive step assumes the statement is true for \( n = k \) and then proves it for \( n = k+1 \).

This process is akin to dominoes falling; once you push the first one (base case), and show one knocks over the next (inductive step), all dominoes fall indefinitely. With the exercise provided, induction is skillfully employed to unfold the identity \( \sin(2^n x) = 2^n \sin x \prod_{j=0}^{n-1} \cos(2^j x) \) for every natural number \( n \). This elegant process is quite powerful in unveiling and proving intricate patterns in mathematics.

Double Angle Identity

The double angle identity is a pivotal trigonometric identity that expresses trigonometric functions of double angles in terms of single angles. The specific double angle identity used in the exercise is \( \sin(2x) = 2 \sin(x) \cos(x) \). This formula demonstrates how doubling an angle reconfigures the sine function's angle representation using both sine and cosine expressions of the original angle.

Some important double angle identities also include:

• \( \cos(2x) = \cos^2(x) - \sin^2(x) \),
• \( \cos(2x) = 1 - 2\sin^2(x) \),
• \( \cos(2x) = 2\cos^2(x) - 1 \).

These identities simplify complex trigonometric functions, aiding in solving equations and verifying identities. In the given problem, the double angle identity for sine helps step by step in generating higher power angles and forming a multi-faceted expression essential for broader algebraic applications.