Questions: If sec α=-5/3, π/2<α<π, then find the exact value of each of the following. a. sin α/2 b. cos α/2 c. tan α/2 a. sin α/2= (Simplify your answer, including any radicals. Use integers or fractions for any numbers)

Solution Steps

To solve this problem, we need to use trigonometric identities and properties. Given that \(\sec \alpha = -\frac{5}{3}\) and \(\frac{\pi}{2} < \alpha < \pi\), we can find \(\cos \alpha\) since \(\sec \alpha = \frac{1}{\cos \alpha}\). Then, use the identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) to find \(\sin \alpha\). For the half-angle identities, use:

• \(\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}}\)
• \(\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}}\)
• \(\tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha}\)

Since \(\frac{\pi}{2} < \alpha < \pi\), \(\alpha\) is in the second quadrant, where \(\sin \alpha > 0\) and \(\cos \alpha < 0\). The sign of the half-angle functions will depend on the quadrant of \(\frac{\alpha}{2}\).

Given that \(\sec \alpha = -\frac{5}{3}\), we find \(\cos \alpha\) as follows: \[ \cos \alpha = \frac{1}{\sec \alpha} = \frac{1}{-\frac{5}{3}} = -\frac{3}{5} = -0.6 \]

Using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\), we can find \(\sin \alpha\): \[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(-\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \] Thus, \[ \sin \alpha = \sqrt{\frac{16}{25}} = \frac{4}{5} = 0.8 \]

Using the half-angle identity: \[ \sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}} = \sqrt{\frac{1 - \left(-\frac{3}{5}\right)}{2}} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{8}{10}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \approx 0.8944 \]

Using the half-angle identity: \[ \cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{\frac{2}{5}}{2}} = \sqrt{\frac{2}{10}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \approx 0.4472 \]

Using the identity: \[ \tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{\frac{4}{5}}{1 - \frac{3}{5}} = \frac{\frac{4}{5}}{\frac{2}{5}} = 2.0 \]

Final Answer