Question

If tan theta = 4/3 and pi

Answer 1

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}`