Question

I need to find the graph. there is multiple options

Answer 1

We have to find θ, where:

`\begin{gathered} \sec (\theta)=(1)/(\cos (\theta))=-(13)/(5) \\ \tan (\theta)<0 \end{gathered}`

We can transform the first equation as:

`\sec (\theta)=-(13)/(5)\Rightarrow\cos (\theta)=(1)/(-(13)/(5))=-(5)/(13)`

As the tangent of the angle is negative, and the tangent is:

`\tan (\theta)=(\sin (\theta))/(\cos (\theta))`

the sine of the angle has to be positive (a quotient between a positive and a negative number is negative).

Then, if the sine is positive and the cosine is negative, then theta is in the second quadrant, between 90° and 180°.

We then can calculate the angle as:

`\cos (\theta)=-(5)/(13)\Rightarrow\theta=\arccos (-(5)/(13))\approx113\degree`

Answer: θ is approximately 113°, located in the second quadrant.