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Chapter 6: Problem 94

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 \(30掳\).

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