Question 667117
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(1) using an angle addition formula....<br>
{{{-5pi/12=(-3pi/12)+(-2pi/12)=(-pi/4)+(-pi/6)}}}<br>
Use {{{sin(A+B)=sin(A)cos(B)+cos(A)sin(B)}}} with {{{A=-pi/4}}} and {{{B=-pi/6}}}<br>
{{{sin(-5pi/12)=sin(-pi/4)cos(-pi/6)+cos(pi/4)sin(-pi/6)}}}
{{{sin(-5pi/12)=(-sqrt(2)/2)(sqrt(3)/2)+(sqrt(2)/2)(-1/2)}}}
{{{sin(-5pi/12)=-sqrt(6)/4-sqrt(2)/4)}}}
{{{sin(-5pi/12)=(-sqrt(6)-sqrt(2))/4}}}<br>
OR...<br>
(2) using an angle subtraction formula...<br>
{{{-5pi/12=(3pi/12)-(8pi/12)=(pi/4)-(2pi/3)}}}<br>
Use {{{sin(A-B)=sin(A)cos(B)-cos(A)sin(B)}}} with {{{A=pi/4}}} and {{{B=2pi/3}}}<br>
{{{sin(-5pi/12)=sin(pi/4)cos(2pi/3)-cos(pi/4)sin(2pi/3)}}}
{{{sin(-5pi/12)=(sqrt(2)/2)(-1/2)-(sqrt(2)/2)(sqrt(3)/2)}}}
{{{sin(-5pi/12)=-sqrt(2)/4-sqrt(6)/4}}}
{{{sin(-5pi/12)=(-sqrt(2)-sqrt(6))/4}}}<br>
ANSWER: {{{sin(-5pi/12)=(-sqrt(6)-sqrt(2))/4}}}<br>