Given that Θ is in the second quadrant
[tex]\cos \theta=-\frac{15}{17}[/tex]
To Determine:
[tex]\sin (\frac{\theta}{2})[/tex]
Solution:
Using Identity
[tex]\sin (\frac{\theta}{2})=\sqrt[]{\frac{1-\cos\theta}{2}}[/tex]
Substitute cos Θ into the formula
[tex]\begin{gathered} \sin (\frac{\theta}{2})=\sqrt[]{\frac{1-\cos\theta}{2}} \\ \sin (\frac{\theta}{2})=\sqrt[]{\frac{1-(-\frac{15}{17})}{2}} \\ \sin (\frac{\theta}{2})=\sqrt[]{\frac{1+\frac{15}{17}}{2}} \end{gathered}[/tex][tex]\sin (\frac{\theta}{2})=\pm\sqrt[]{\frac{\frac{17+15}{17}}{2}}=\pm\sqrt[]{\frac{\frac{32}{17}}{2}}=\pm\sqrt[]{\frac{32}{17}\times\frac{1}{2}}=\pm\sqrt[]{\frac{16}{17}}[/tex][tex]\begin{gathered} \sin (\frac{\theta}{2})=\frac{\pm4}{\sqrt[]{17}}=\frac{\pm4}{\sqrt[]{17}}\times\frac{\sqrt[]{17}}{\sqrt[]{17}} \\ \sin (\frac{\theta}{2})=\frac{\pm4\sqrt[]{17}}{17} \end{gathered}[/tex]
Hence, the final answer is
[tex]\sin (\frac{\theta}{2})=\frac{\pm4\sqrt[]{17}}{17}[/tex]
