Question 190623
Here's the unit circle for reference:


<img src="http://www.math.ucsd.edu/%7Ejarmel/math4c/Unit_Circle_Angles.png">




{{{sin(-10pi/3)}}} Start with the given expression.



{{{-sin(10pi/3)}}} Use the identity {{{sin(-x)=-sin(x)}}}



Take note that {{{10pi/3=2pi+4pi/3}}}



{{{-sin(2pi+4pi/3)}}} Break up {{{10pi/3}}} to get {{{2pi+4pi/3}}}



{{{-(sin(2pi)cos(4pi/3)+cos(2pi)sin(4pi/3))}}} Use the identity {{{sin(x+y)=sin(x)cos(y)+cos(x)sin(y)}}} to expand



{{{-((0)cos(4pi/3)+(1)sin(4pi/3))}}} Take the sine of {{{2pi}}} to get 0. Take the cosine of {{{2pi}}} to get 1 (use the unit circle).



{{{-(0+sin(4pi/3))}}} Multiply



{{{-sin(4pi/3)}}} Simplify




Take note that {{{4pi/3=pi+pi/3}}}




{{{-sin(pi+pi/3)}}} Break up {{{4pi/3}}} to get {{{pi+pi/3}}}



{{{-(sin(pi)cos(pi/3)+cos(pi)sin(pi/3))}}} Use the identity {{{sin(x+y)=sin(x)cos(y)+cos(x)sin(y)}}} to expand



{{{-((0)cos(pi/3)+(-1)sin(pi/3))}}} Take the sine of {{{pi}}} to get 0. Take the cosine of {{{pi}}} to get -1 (use the unit circle).



{{{-(0-sin(pi/3))}}} Multiply



{{{sin(pi/3)}}} Simplify



{{{sqrt(3)/2}}} Evaluate the sine of {{{pi/3}}} to get {{{sqrt(3)/2}}} (use the unit circle).




So {{{sin(-10pi/3)=sqrt(3)/2}}}