Problem 96 Use the identity $$\sin 2 x=2 ... [FREE SOLUTION]

Chapter 8: Problem 96

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

Short Answer

Expert verified

Use the identity iteratively and apply induction to show the formula is valid for any $ n $.

Step by step solution

Introduction of the Problem

We need to use the trigonometric identity $ \sin 2x = 2 \sin x \cos x $ iteratively to express $ \sin(2^n x) $. The goal is to express that in a product form involving cosine terms.

Apply the Identity Once

According to the given identity, $ \sin(2x) = 2 \sin x \cos x $. We'll start by applying this formula to $ \sin(2x) $.Therefore, the expression becomes $ \sin(2x) = 2 \sin x \cos x $.

Second Application

Apply the identity to $ \sin(4x) = \sin(2 \times 2x) $. Here, $ x $ is $2x$, and we get:\[ \sin(4x) = 2 \sin(2x) \cos(2x) \]Substitute the result from Step 1:\[ \sin(4x) = 2 (2 \sin x \cos x) \cos(2x) = 2^2 \sin x \cos x \cos 2x \]

Apply Inductive Step

Assume $ \sin(2^k x) = 2^k \sin x \cos x \cos 2x \cdots \cos 2^{k-1}x $. For $ k+1 $, apply the identity:\[ \sin(2^{k+1}x) = 2 \sin(2^k x) \cos(2^k x) \]Use the inductive hypothesis: \[ \sin(2^{k+1}x) = 2 (2^k \sin x \cos x \cos 2x \cdots \cos 2^{k-1}x) \cos(2^k x) \]Thus,\[ \sin(2^{k+1}x) = 2^{k+1} \sin x \cos x \cos 2x \cdots \cos 2^k x \]

Conclusion

By applying the base case and the inductive step repeated $ n $ times, we obtain:\[ \sin(2^n x) = 2^n \sin x \cos x \cos 2x \cdots \cos 2^{n-1} x \]Hence, the identity is proven by induction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Induction

Mathematical induction is a powerful technique used to prove statements about integers. It's much like climbing a ladder, where you prove one step at a time.

First, the **base case** is proven. This is usually the simplest version of the problem, such as the first rung of the ladder. In our trigonometric example, this base case involves showing the identity holds for a small number like 2.
Next, we assume the statement holds for some arbitrary step, say at step $ k $. This step is called the **inductive hypothesis**. It鈥檚 like believing you can hold onto the ladder at step $ k $.
Finally, you must prove the statement is true for step $ k+1 $. If you do this, then by the principle of induction, the statement must be true for all successive steps 鈥� from the base case onwards.

This way, induction ensures a logical flow from one known truth to another. In trigonometry, induction helps extend a known identity through multiples, like proving more extended identities from simpler ones.

Sine and Cosine Functions

Sine and cosine are foundational functions in trigonometry that describe oscillating waves and are seen everywhere from the swinging of a pendulum to the alternating current in electricity.

**Sine Function, $ \sin x $**: This function gives the vertical coordinate of a point on a unit circle corresponding to an angle $ x $. It's periodic with a cycle of $ 2\pi $ radians.
**Cosine Function, $ \cos x $**: Similarly, this function provides the horizontal coordinate, and it also has a period of $ 2\pi $.
In our problem, these functions are manipulated via an identity $ \sin 2x = 2 \sin x \cos x $, which expresses a double-angle formula for sine.

These identities and properties of sine and cosine are keys to simplifying and solving complex trigonometric problems and are often used interchangeably to derive new relations.

Mathematical Proofs

Mathematical proofs provide the backbone of mathematics by logically showing that a statement or theorem is precisely accurate.

Proofs can be combinatorial, geometric, or algebraic, and each has its methodology.
In trigonometry, proofs often rely on algebraic manipulation of identities, as seen in our exercise, using $ \sin(2x) = 2 \sin x \cos x $.
A good proof is usually clear, concise, and uses logical steps that build on each other to demonstrate a statement鈥檚 truth.

To assist with understanding, proofs often break down complex expressions into smaller, more manageable parts, as demonstrated in the given exercise's step-by-step solution. Thus, understanding how to build and verify proofs is essential for success in mathematics.

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91影视

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

Short Answer

Step by step solution

Introduction of the Problem

Apply the Identity Once

Second Application

Apply Inductive Step

Conclusion

Key Concepts

Induction

Sine and Cosine Functions

Mathematical Proofs

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