Question

If θ is a first quadrant angle in standard position with P(u,v) = (3,4) evaluate tan 1/2 θ1) 1/42) 1/23) 2/3

Answer 1

Given

If θ is a first quadrant angle in standard position with P(u,v) = (3,4) .

Answer

`\begin{gathered} Tan\theta=(4)/(3) \\ \sec ^2\theta=1+\tan ^2\theta \\ \sec \theta=\sqrt[]{1+((4)/(3))}^2 \\ \sec \theta=\pm(5)/(3) \\ \cos \theta=(3)/(5) \\ \sin \theta=\sqrt[]{1-\cos ^2\theta} \\ =\sqrt[]{1-(9)/(25)} \\ =\sqrt[]{(16)/(25)}=(4)/(5) \end{gathered}`

Using identity

`\begin{gathered} \tan (1)/(2)\theta=(1-\cos \theta)/(\sin \theta) \\ =(1-(3)/(5))/((4)/(5))=((2)/(5))/((4)/(5)) \\ =(1)/(2) \end{gathered}`