Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

So I have this question: $\cos X = -5/13$ where $0<X<2\pi$ (less then or equal signs not just less then). Determine all possible trig ratios. I want to know how you would find the trig ratios.

The answers at the back of the textbook are $$\eqalign{\sin X &= 12/13 {\rm\ or\ -12/13}\cr \tan X& = 12/5 {\rm\ or\ } -12/5\cr \sec X &= -13/5.}$$

But I don't get how they got those answers.

share|improve this question
add comment

2 Answers

up vote 1 down vote accepted

Draw a right triangle in the second quadrant that looks like this: $\ \ \ \ \lower6pt{ \llap{opp\ }|\nwarrow \rlap{\ hyp}\over adj}$

Since $\cos X=-5/13$, the hypotenuse, $hyp$, of that triangle has length 13 and the horizontal leg, $adj$, has length 5.

The Pythagorean Theorem will tell you that the length of the vertical side, $opp$, of the triangle is $\sqrt{169-25}=\sqrt{144}=12$.

So our triangle is $\ \ \ \ \lower6pt{ \llap{12\ }|\nwarrow \rlap{\ 13}\over 5}$

Now read the trig ratios from the triangle, attaching the appropriate sign. For example, $\tan X= {\rm opp.\over adj}=-12/5$. The negative sign is needed since $\tan$ is negative in the second quadrant. The other trig ratios I'll leave to you.

You also need to do the above for a triangle in the third quadrant (since $\cos$ can be negative there).

share|improve this answer
add comment

You have $\sin^2 X + \cos^2 X=1$, so $\sin^2 X = 1 - \frac{25}{169} = \frac{144}{169}, \sin X = \pm \frac{12}{13}$. Then $\tan X = \frac{\sin X}{\cos X}$ and you have to check which signs are applicable.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.