# Answer to Question #19793 in Trigonometry for kevin

Question #19793

Find the exact value by using a half-angle identity.

sin 75 degrees

sin 75 degrees

Expert's answer

we know that sin(90-x)=cos(x)

also we know that cos(2x)=2cos^2(x)-1 socos^2(x)=(1+cos(2x))/2 and for sin(75)=sin(90-15)=cos(15) but ( cos(15)

)^2=(1+cos(2*15))/2=(1+cos(30))/2 sin(75)=cos(15)=sqrt( (1+cos(30))/2)=sqrt(

(1+sqrt(3)/2 )/2)=sqrt( [2+sqrt(3)]/4 )=sqrt(2+sqrt(3) )/2

also we know that cos(2x)=2cos^2(x)-1 socos^2(x)=(1+cos(2x))/2 and for sin(75)=sin(90-15)=cos(15) but ( cos(15)

)^2=(1+cos(2*15))/2=(1+cos(30))/2 sin(75)=cos(15)=sqrt( (1+cos(30))/2)=sqrt(

(1+sqrt(3)/2 )/2)=sqrt( [2+sqrt(3)]/4 )=sqrt(2+sqrt(3) )/2

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