Question 1071585
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I need assistance in finding the sin(x/2), cos(x/2), and tan(x/2) from: {{{sec(x) = 4/3}}} with 270 degrees < x < 360 degrees
sin(x/2) = 
cos (x/2) = 
tan (x/2) = 

Step by step solutions is most appreciated. Thank you.
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<pre>
0.  Since sec(x) = {{{4/3}}}, it implies that cos(x) = {{{3/4}}}.


1.  Now, according to the formulas for the half argument,

    sin(x/2) = {{{+sqrt((1-cos(x))/2)}}} = {{{+sqrt((1-(3/4))/2)}}} = {{{+sqrt((1/4)/2)}}} = {{{sqrt(1/8)}}} = {{{1/(2*sqrt(2))}}} = {{{sqrt(2)/4}}}. 

    Notice that the sign at sqrt is "+" (plus), because the angle {{{x/2}}} lies in QII, where sine is positive. 



2.  Next, according to the formulas for the half argument,

    cos(x/2) = {{{-sqrt((1+cos(x))/2)}}} = {{{-sqrt((1+(3/4))/2)}}} = {{{-sqrt((7/4)/2))}}} = {{{-sqrt(7/4)}}} = {{{-sqrt(7)/2}}}.

    Notice that the sign at sqrt is "-" (minus), because the angle {{{x/2}}} lies in QII, where cosine is negative. 


3.  Just having  sin(x/2) = {{{sqrt(2)/4}}}  and  cos(x/2) = {{{-sqrt(7)/2}}},  you can calculate tan(x/2)  as their ratio:

    tan(x/2) = {{{((sqrt(2)/4))/((-sqrt(7)/2))}}} = {{{-sqrt(2)/(2*sqrt(7))}}} = {{{-sqrt(14)/14)}}}
</pre>

Regarding formulas for trigonometric functions of half argument, &nbsp;see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Compendium-of-Trigonometry-Formulas.lesson>FORMULAS FOR TRIGONOMETRIC FUNCTIONS</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF = http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>

in this site.


Also, &nbsp;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. Formulas for trigonometric functions</U>".


Also see the topic "<U>Trigonometry: Solved problems</A>" of this textbook.