# Finding exact values of trig functions

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?

• are the identities such as $\tan^x+1=\sec^2x$ open to us? i.e. have you come across this? Aug 4, 2014 at 22:25
• @Chinny84 No, I don't think so, I haven't learned that Aug 4, 2014 at 22:26
• 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 $\sin\theta=-\frac{1}{\sqrt{26}}$. Aug 4, 2014 at 22:30
• 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. Aug 4, 2014 at 22:32
• @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. Aug 4, 2014 at 23:00

$\tan \theta = \cfrac {\sin \theta}{\cos\theta} = -\cfrac 15$
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 $26\cos^2 \theta =25$