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If tan theta = 4/3 and pi

User Matt Sich
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Given that tan theta = 4/3 and theta lies in the third quadrant.


\pi<\theta<(3\pi)/(2)

Divide the compound inequality by 2.


(\pi)/(2)<(\theta)/(2)<(3\pi)/(4)

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.


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

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,


\begin{gathered} \cos ((\theta)/(2))=-\sqrt[]{(1+\cos x)/(2)} \\ =-\sqrt[]{(1-(3)/(5))/(2)} \\ =-\sqrt[]{((2)/(3))/(2)} \\ =-\sqrt[]{(1)/(3)} \end{gathered}

User Dice
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