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Chapter 20: Problem 16

Simplify the given expressions. $$\sqrt{2-2 \cos 64 x}$$

Short Answer

Expert verified
The simplified expression is \( 2|\sin(32x)| \).

Step by step solution

01

Identify the Trigonometric Identity

To simplify the expression \( \sqrt{2 - 2 \cos 64x} \), we start by recalling a trigonometric identity. The expression inside the square root can be associated with the double angle identity for sine: \[ 2 - 2\cos(\theta) = 4\sin^2\left(\frac{\theta}{2}\right) \].
02

Apply the Trigonometric Identity

Using the identity from Step 1, rewrite the expression inside the square root:\[ \sqrt{2 - 2\cos 64x} = \sqrt{4\sin^2\left(\frac{64x}{2}\right)} \] which simplifies to \[ \sqrt{4\sin^2(32x)} \].
03

Simplify the Square Root

Simplify the square root of the expression:\[ \sqrt{4\sin^2(32x)} = \sqrt{4} \cdot \sqrt{\sin^2(32x)} = 2 \cdot |\sin(32x)| \]. To maintain the principal value of the square root, we use the absolute value.

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Key Concepts

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

Trigonometric Identity
Trigonometric identities are crucial tools in simplifying trigonometry-based expressions. They help us relate different trigonometric functions and make complex expressions manageable. In this exercise, we use the identity related to the sine of double angles. The identity is given by:
  • \(2 - 2\cos(\theta) = 4\sin^2\left(\frac{\theta}{2}\right)\)
This identity is derived from the cosine double-angle identity \(\cos(2\alpha) = 1 - 2\sin^2(\alpha)\), by rearranging terms. It helps us transform the difficult cosine expression into a simpler sine-related form. Understanding these relationships is key in mastering trigonometry, as they serve as shortcuts in simplifying expressions.
Double Angle Identity
The concept of double angle identity arises when you think about an angle that is double another angle, hence the name. The double angle identity for sine, in particular, can be quite useful in various scenarios. However, the identity we used here was not directly for a double angle but was rearranged to involve sine. Here's how it works:
  • The basic double angle identity for cosine is \(\cos(2\theta) = 1 - 2\sin^2(\theta)\).
  • We adjusted and used it in the form of \(2 - 2\cos(64x) = 4\sin^2\left(\frac{64x}{2}\right)\).
By using these identities, we simplify complex expressions by bringing them into a form that involves simpler trigonometric operations, often reducing calculations.
Simplifying Expressions
Simplifying expressions is a fundamental skill in mathematics which aims to make expressions less cumbersome and more elegant. In this problem, simplification was approached through a series of algebraic and trigonometric manipulations.Let's break down the process:
  • Start with the complex expression \(\sqrt{2 - 2\cos(64x)}\).
  • Recognize, through identities, that it can be rewritten using sine terms: \(\sqrt{4\sin^2(32x)}\).
  • Further simplify by taking the square root of each factor, leading to \(2 \cdot |\sin(32x)|\).
When simplifying, it's crucial to consider absolute values to maintain the principal value of square roots, ensuring the expressions remain valid under all circumstances. The result is a simplified form that is both easier to understand and compute.

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Most popular questions from this chapter

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