Find the exact value of $\sec \theta$ given the value of $\tan \theta$
If $\tan \theta = 7$ and $-\pi < \theta < 0$ then what is the exact value of $\sec \theta?$
I tried doing $\tan \theta = \frac{y}{x}= 7 $, so $y = 7x$ and subbed that into $1 = \sqrt{y^2 + x^2}$, however I ended up getting $x = \frac{1}{5\sqrt{2}}$ and thus $\sec \theta = -5\sqrt{2}$ however this is incorrect.
EDIT: sorry I accidentally left out the negative sign and I did indeed consider the quadrants, I actually got the right answer but I stupidly thought I didn't since the correct answer was listed as $-\frac{10}{\sqrt{2}}$. For some reason it did not occur to me to rationalize it. However thanks for your responses since when I saw that you guys had the same answer as me it made me go back and look at the "correct" answer listed, so now I realized I simply overlooked something really silly.
$\endgroup$ 13 Answers
$\begingroup$You can use the identity $1+\tan^2\theta=\sec^2{\theta}$.
Since you know that $\displaystyle -\pi < \theta < -\frac{\pi}{2}$ from the sign of $\tan$, you can determine that $\sec$ is negative.
Now, you have that $\sec(\theta)=-\sqrt{50}=-5\sqrt{2}$.
$\endgroup$ $\begingroup$Since $\tan\theta >0 $, the angle $\theta$ must be in the third quadrant, over which secant function takes negative values.
Thus,
$$\sec\theta =- \sqrt{1+\tan^2\theta}=- \sqrt{1+49}= -5\sqrt 2$$
$\endgroup$ $\begingroup$The idea here is that if $\tan\theta=7$, $\theta\in(-\pi,0)$, this tells us that the angle is in the third quadrant, and as such, $\sec\theta$ has to be negative. You are on the right track, in general, but the correct answer would be $\boxed{\sec\theta=-5\sqrt2}$.
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