Question 1204691
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Hint #1
When 90 < x < 180, {{{csc(x) = 3}}} leads to {{{cos(x) = -2*sqrt(2)/3}}}



Hint #2
90 < x < 180 has all sides cut in half to 45 < x/2 < 90
Angle x/2 is in quadrant 1 where all 6 trig functions are positive



Hint #3
The half angle identities are
{{{sin(x/2) = ""+-sqrt( (1-cos(x))/2 )}}}


{{{cos(x/2) = ""+-sqrt( (1+cos(x))/2 )}}}


{{{tan(x/2) = ""+-sqrt( (1-cos(x))/(1+cos(x)) )}}}


However, keep the 2nd hint in mind, so we can drop the plus minus to write
{{{matrix(1,3,sin(x/2) = sqrt( (1-cos(x))/2 ), " when ", 45 < x/2 < 90)}}}


{{{matrix(1,3,cos(x/2) = sqrt( (1+cos(x))/2 ), " when ", 45 < x/2 < 90)}}}


{{{matrix(1,3,tan(x/2) = sqrt( (1-cos(x))/(1+cos(x)) ), " when ", 45 < x/2 < 90)}}}


Or once you know what sin(x/2) and cos(x/2) are, you can compute tangent like so
{{{tan(x/2) = (sin(x/2))/(cos(x/2))}}}


More trig identities can be found here
<a href="https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf">https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf</a>
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