Given
If θ is a first quadrant angle in standard position with P(u,v) = (3,4) .
Answer
[tex]\begin{gathered} Tan\theta=\frac{4}{3} \\ \sec ^2\theta=1+\tan ^2\theta \\ \sec \theta=\sqrt[]{1+(\frac{4}{3})}^2 \\ \sec \theta=\pm\frac{5}{3} \\ \cos \theta=\frac{3}{5} \\ \sin \theta=\sqrt[]{1-\cos ^2\theta} \\ =\sqrt[]{1-\frac{9}{25}} \\ =\sqrt[]{\frac{16}{25}}=\frac{4}{5} \end{gathered}[/tex]Using identity
[tex]\begin{gathered} \tan \frac{1}{2}\theta=\frac{1-\cos \theta}{\sin \theta} \\ =\frac{1-\frac{3}{5}}{\frac{4}{5}}=\frac{\frac{2}{5}}{\frac{4}{5}} \\ =\frac{1}{2} \end{gathered}[/tex]