Because the tan (theta) = -2, and sin (theta) < 0, we can use the following to determine the value of theta:
[tex]\begin{gathered} \tan (\theta)=\frac{\sin(\theta)}{\cos(\theta)}=-2 \\ \sin (\theta)<0\Rightarrow\cos (\theta)>0 \end{gathered}[/tex]We can use the following relationship to develop it:
[tex]\begin{gathered} \tan ^2(\theta)+1=\sec ^2(\theta) \\ \sec ^2(\theta)=1+(-2)^2=5 \\ \sec (\theta)=\pm\sqrt[]{5}^{} \end{gathered}[/tex]but, because cos(theta) > 0, we have:
[tex]\begin{gathered} \cos (\theta)>0 \\ \sec (\theta)=\frac{1}{\cos(\theta)}\Rightarrow\sec (\theta)>0 \\ \sec (\theta)=\sqrt[]{5} \end{gathered}[/tex]