given tan s < 0, sec s = 11/4, find sin s

[Sonic]

I am having trouble understanding a problem in my Trigonometry course. In this problem, here is the information that I am given already:

sec s = 11/4
tan s < 0

I need to find sin s. I am not sure how I can find sin s when tangent is not given. Tangent is less than zero does not tell me anything.

 

[morson]

Yes it does, it tells you that the angle lies only in either the 2nd or 4th quadrant.

The fact that sec(s) = 1/cos(s) = 11/4 tells you that cos(s) is positive and so therefore the angle is in the 4th quadrant.

Since cos(s) is positive, cos(s) is the positive square root of 1 - sin^2(s). Also note that sin(s) is negative, so take the negative square root when solving for sin(s).

 

[Sonic]

What trigonometric identity do I use? I am lost.

 

[pka]

In this problem, here is the information that I am given already:
sec s = 11/4 & tan s < 0.

These two combined tell us that s is in the fourth quadrant.
We also know that \(\displaystyle \L \cos (s) = \frac{4}{{11}}\).
Recall that \(\displaystyle \L
\begin{array}{l}
\cos (t) = \frac{x}{r},\quad \sin (t) = \frac{y}{r}\quad \& \quad r = \sqrt {x^2 + y^2 } \\
\end{array}.\)

Now you have all the tools; solve the problem.

 

[Sonic]

I know that but I am not sure what trigonometric identity to use. The answer in the back of the book is -√105 / 11. I have no idea how they got that answer.

 

[skeeter]

I know that but I am not sure what trigonometric identity to use. The answer in the back of the book is -√105 / 11. I have no idea how they got that answer.

this basic identity look familiar?

\(\displaystyle \L \cos^2(s) + \sin^2(s) = 1\)

s is in quad IV ... cos(s) = 4/11, sin(s) < 0. solve for sin(s)

 

[Sonic]

I figured out that I just did not square cosine. I came up with the right answer this time. Thanks guys.