Question

If tan θ = -2 and sin θ < 0, find sec θ.

Answer 1

Because the tan (theta) = -2, and sin (theta) < 0, we can use the following to determine the value of theta:

`\begin{gathered} \tan (\theta)=(\sin(\theta))/(\cos(\theta))=-2 \\ \sin (\theta)<0\Rightarrow\cos (\theta)>0 \end{gathered}`

We can use the following relationship to develop it:

`\begin{gathered} \tan ^2(\theta)+1=\sec ^2(\theta) \\ \sec ^2(\theta)=1+(-2)^2=5 \\ \sec (\theta)=\pm\sqrt[]{5}^{} \end{gathered}`

but, because cos(theta) > 0, we have:

`\begin{gathered} \cos (\theta)>0 \\ \sec (\theta)=(1)/(\cos(\theta))\Rightarrow\sec (\theta)>0 \\ \sec (\theta)=\sqrt[]{5} \end{gathered}`