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?
Best Answer
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.