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Chapter 6: Problem 74

Show that $$ |\sin (2 \theta)| \leq 2|\sin \theta| $$ for every angle \(\theta\).

Short Answer

Expert verified
We used the double angle formula \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) and properties of absolute values to show that \(|\sin(2\theta)| \leq 2|\sin(\theta)|\). By simplifying the inequality, we found out that this inequality holds if \(|\cos(\theta)| \leq 1\), which is always true for every angle \(\theta\). Therefore, \(|\sin(2\theta)| \leq 2|\sin(\theta)|\) for every angle \(\theta\).

Step by step solution

01

Write given inequality with double angle formula

First, let's rewrite \(|\sin(2\theta)|\) using the double angle formula for sine: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). So, the given inequality becomes: $$ |2\sin(\theta)\cos(\theta)| \leq 2|\sin(\theta)| $$
02

Simplify inequality using the absolute value product rule

Using the rule \(|ab| = |a|\times|b|\), we can rewrite the left side of the inequality: $$ |2|\times|\sin(\theta)|\times|\cos(\theta)| \leq 2|\sin(\theta)| $$
03

Divide both sides by \(2|\sin(\theta)|\)

As long as \(\sin(\theta) \neq 0\), we can safely divide both sides by \(2|\sin(\theta)|\): $$ |\cos(\theta)| \leq 1 $$
04

Conclude the proof

Considering the properties of the cosine function, we know that the maximum and minimum values of cosine are 1 and -1, respectively. Therefore, the absolute value of the cosine function is always less than or equal to 1: $$ |\cos(\theta)| \leq 1 $$ Thus, we have shown that \(|\sin(2\theta)| \leq 2|\sin(\theta)|\) for every angle \(\theta\).

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Most popular questions from this chapter

Find angles \(u\) and \(v\) such that \(\sin (2 u)=\) \(\sin (2 v)\) but \(|\sin u| \neq|\sin v|\)

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.3 .\) (a) Without using a double-angle formula, evaluate \(\cos (2 \theta)\). (b) Without using an inverse trigonometric function, evaluate \(\cos (2 \theta)\) again.

Suppose \(\theta\) is an angle such that \(\cos \theta\) is rational. Explain why \(\cos (2 \theta)\) is rational.

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