Question 1076776
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If csc (theta) <0 and cos(theta)=5/13, find sin (theta/2) and cos (theta/2).
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<pre>
{{{csc(theta)}}} < {{{0}}} means {{{sin(theta)}}} < {{{0}}}.

Together with {{{cos(theta)}}} = {{{5/13}}}, it means that {{{theta}}} is the angle in QIV.

Hence,  {{{theta/2}}} is the angle in QII.


Then  {{{sin(theta/2)}}} = {{{sqrt((1-cos(theta))/2)}}} = {{{sqrt((1-(5/13))/2)}}} = {{{sqrt((13-5)/(13*2))}}} = {{{sqrt(8/26)}}} = {{{2/sqrt(13)}}} = {{{(2*sqrt(13))/13}}}.

The sign is "+" since sine is positive in QII.


{{{cos(theta/2)}}} = {{{-sqrt((1+cos(theta))/2)}}} = {{{-sqrt((1+(5/13))/2)}}} = {{{-sqrt((13+5)/(13*2))}}} = {{{-sqrt(18/26)}}} = {{{-sqrt(9/13)}}} = {{{(-3)/sqrt(13)}}}.

The sign is "-" since cosine is negative in QII.
</pre>

To see more solved similar problems on calculating trig functions look into the lessons  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Calculating-trigonometric-functions-of-angles.lesson>Calculating trigonometric functions of angles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Selected-problems-from-the-archive-on-calculating-trig-functions-of-angles.lesson>Advanced problems on calculating trigonometric functions of angles</A>

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Trigonometry: Solved problems</U>".