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Sagot :
Given that tan theta = 4/3 and theta lies in the third quadrant.
[tex]\pi<\theta<\frac{3\pi}{2}[/tex]Divide the compound inequality by 2.
[tex]\frac{\pi}{2}<\frac{\theta}{2}<\frac{3\pi}{4}[/tex]This means theta/2 lies in the second quadrant. So, cos theta/2 and sec theta/2 are negative.
Use trigonometric identities to find sec theta.
[tex]\begin{gathered} \sec \theta=\sqrt[]{1+\tan ^2\theta} \\ =\sqrt[]{1+(\frac{4}{3})^2} \\ =\sqrt[]{1+\frac{16}{9}} \\ =\sqrt[]{\frac{25}{9}} \\ =-\frac{5}{3} \end{gathered}[/tex]we know that cosine is the inverse of secant. So, cos theta = -3/5.
now, using the half-angle formula, we have to find cos theta/2,
[tex]\begin{gathered} \cos (\frac{\theta}{2})=-\sqrt[]{\frac{1+\cos x}{2}} \\ =-\sqrt[]{\frac{1-\frac{3}{5}}{2}} \\ =-\sqrt[]{\frac{\frac{2}{3}}{2}} \\ =-\sqrt[]{\frac{1}{3}} \end{gathered}[/tex]
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