Chapter 1: Problem 2

Determine the exact values of a. \(\sin \frac{\pi}{8}\). b. \(\tan 15^{\circ}\). c. \(\cos 105^{\circ}\).

Short Answer

Expert verified

The exact values of the trigonometric functions are (a) \(\sin \frac{\pi}{8} = \frac{\sqrt{2 - \sqrt{2}}}{2}\), (b) \(\tan 15^{\circ} = 2 - \sqrt{3}\), and (c) \(\cos 105^{\circ} = -\frac{\sqrt{6} + \sqrt{2}}{4}\)

Step by step solution

Determine the exact value of \(\sin \frac{\pi}{8}\)

To find the exact value of \(\sin \frac{\pi}{8}\), we will use the half angle identity \(\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}\). Here, we have \(\alpha = \frac{\pi}{4}\), because \(\frac{\pi}{8} = \frac{1}{2} \frac{\pi}{4}\). So, \(\sin \frac{\pi}{8} = \sin \frac{\frac{\pi}{4}}{2} = \sqrt{\frac{1 - \cos \frac{\pi}{4}}{2}} = \sqrt{\frac{1 - \frac{1}{\sqrt{2}}}{2}}\). Which simplifies to \(\frac{\sqrt{2 - \sqrt{2}}}{2}\)

Determine the exact value of \(\tan 15^{\circ}\)

To find this value, use the difference of angles formula: \(\tan (a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\). Let \(a = 45^{\circ}\) and \(b = 30^{\circ}\), since \(45^{\circ} - 30^{\circ} = 15^{\circ}\). So, \(\tan 15^{\circ} = \tan (45^{\circ} - 30^{\circ}) = \frac{\tan 45^{\circ} - \tan 30^{\circ}}{1 + \tan 45^{\circ} \tan 30^{\circ}} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = 2 - \sqrt{3}\)

Determine the exact value of \(\cos 105^{\circ}\)

To find this value, use the sum of angles formula: \(\cos (a + b) = \cos a \cos b - \sin a \sin b\). Let \(a = 45^{\circ}\) and \(b = 60^{\circ}\) since \(45^{\circ}+ 60^{\circ} = 105^{\circ}\). Therefore, \(\cos 105^{\circ} = \cos (45^{\circ} + 60^{\circ}) = \cos 45^{\circ} \cos 60^{\circ} - \sin 45^{\circ} \sin 60^{\circ} = \frac{1}{\sqrt{2}} \cdot \frac{1}{2} - \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} = -\frac{\sqrt{6} + \sqrt{2}}{4}\)