/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 95 Give an example of an angle \(\t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 95

Give an example of an angle \(\theta\) such that \(\sin \theta\) is rational but \(\sin (2 \theta)\) is irrational.

Short Answer

Expert verified
An example of an angle \(\theta\) such that \(\sin\theta\) is rational but \(\sin(2\theta)\) is irrational is \(\theta = \frac{\pi}{6}\) or \(30掳\). In this case, \(\sin\theta = \frac{1}{2}\) and \(\sin(2\theta) = \frac{\sqrt{3}}{2}\), which is irrational.

Step by step solution

01

Choose a rational value for \(\sin\theta\)

Consider an angle \(\theta\) such that \(\sin\theta = \frac{1}{2}\). There is a known angle with this property, which is \(\theta = \frac{\pi}{6}\) or \(30掳\).
02

Calculate \(\cos \theta\)

For \(\sin\theta =\frac{1}{2}\), we have the following equation: \[\cos^2 \theta = 1 - \sin^2 \theta\] Substitute the value of \(\sin\theta\): \[\cos^2\theta = 1 - \frac{1}{4}=\frac{3}{4}\] Taking the positive square root of both sides (since the cosine of the first quadrant where \(\frac{\pi}{6}\) is positive): \[\cos \theta = \frac{\sqrt{3}}{2}\]
03

Calculate \(\sin (2\theta)\)

Use the double-angle formula: \[\sin(2\theta) = 2\sin\theta\cos\theta\] Substitute the values of \(\sin\theta\) and \(\cos\theta\): \[\sin(2\theta) = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}\]
04

Verify if \(\sin (2\theta)\) is irrational

The result we obtained for \(\sin(2\theta)\) is \(\frac{\sqrt{3}}{2}\). Since \(\sqrt{3}\) is an irrational number (it cannot be expressed as a fraction), \(\frac{\sqrt{3}}{2}\) is also irrational. So, an example of an angle \(\theta\) such that \(\sin\theta\) is rational but \(\sin(2\theta)\) is irrational is \(\theta = \frac{\pi}{6}\) or \(30掳\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and \(\tan u=-\frac{1}{7} \quad\) and \(\quad \tan v=-\frac{1}{8}\) $$\cos (2 v)$$

Show that $$(\cos x+\sin x)^{2}=1+\sin (2 x)$$ for every number \(x\).

The next two exercises emphasize that \(\sin (x-y)\) does not equal \(\sin x-\sin y .\) For \(x=79^{\circ}\) and \(y=33^{\circ}\), evaluate each of the following: (a) \(\sin (x-y)\) (b) \(\sin x-\sin y\)

Find a formula for \(\tan \left(\theta+\frac{\pi}{2}\right)\).

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

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.