Find exact value of each trigonometric function of $\theta$ if $\tan\theta=-1/5$ and $\sec \theta >0$

I know that $\cot \theta=-5,$ right?

Secant and cosine are positive in the fourth quadrant. I drew a triangle there, with $\theta = \tan^{-1}-1/5$.

The hypotenuse I found to be $5.1.$

So I got these values:

$\sin \theta = -10/51, \cos \theta = 50/51, \sec \theta = 51/50, \csc \theta = -51/10$

Is this correct?

  • $\begingroup$ are the identities such as $\tan^x+1=\sec^2x$ open to us? i.e. have you come across this? $\endgroup$
    – Chinny84
    Aug 4, 2014 at 22:25
  • $\begingroup$ @Chinny84 No, I don't think so, I haven't learned that $\endgroup$
    – Emi Matro
    Aug 4, 2014 at 22:26
  • 2
    $\begingroup$ Calculate $\sin\theta$. It will be negative. Draw a right triangle with legs $1$ (opposite) and $5$ (adjacent). So the hypotenuse is $\sqrt{26}$, and $\sin\theta=-\frac{1}{\sqrt{26}}$. $\endgroup$ Aug 4, 2014 at 22:30
  • $\begingroup$ Going back in time (a lot ) finding exact values required using either the unit triangle for determine angles of 45 60 30 but other angles we used identities like the one above :). So all in all this is a great little question. $\endgroup$
    – Chinny84
    Aug 4, 2014 at 22:32
  • 1
    $\begingroup$ @user437158: Your answers are not far from the truth, and the idea of the computation is fine. However, you may be expected to give "exact" answers. The answers you gave might not be quite adequate for the purposes, for example, of astronomy. $\endgroup$ Aug 4, 2014 at 23:00

1 Answer 1


Here is how I would tackle it, without computing any sides of a triangle:

$\tan \theta = \cfrac {\sin \theta}{\cos\theta} = -\cfrac 15$

So that $\cos \theta = -5 \sin \theta$ and we can use this because we know that $\cos^2\theta+\sin^2\theta = 1$ and substituting for $\cos \theta$ we obtain $26 \sin^2 \theta=1$ and then $26\cos^2 \theta =25$

It remains to use the information you have to identify the relevant signs for the functions.

Your answers are approximately correct, but you should make it clear that they are approximations - giving fractions as you have suggests exactness.


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