Question

If cos(theta) = 4/9 and pi < theta < 2pi. Find the tangent of theta

Answer 1

In order to calculate the tangent of theta, let's first calculate the sine of theta using the following relation:

`\begin{gathered} \sin ^2(\theta)+\cos ^2(\theta)=1 \\ \sin ^2(\theta)+((4)/(9))^2=1 \\ \sin ^2(\theta)+(16)/(81)=1 \\ \sin ^2(\theta)=(81)/(81)-(16)/(81) \\ \sin ^2(\theta)=(65)/(81) \\ \sin (\theta)=\pm_{}\frac{\sqrt[]{65}}{9} \end{gathered}`

Since pi < theta < 2pi, the sine is negative.

Now, calculating the tangent, we have:

`\begin{gathered} \tan (\theta)=(\sin (\theta))/(\cos (\theta)) \\ \tan (\theta)=\frac{-\frac{\sqrt[]{65}}{9}}{(4)/(9)} \\ \tan (\theta)=-\frac{\sqrt[]{65}}{4} \end{gathered}`