How do you use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle #75^circ#?

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Answer 1 · 2017-07-21T18:02:52

See the explanation below

Conversion of #75^@# into radians

#=75/180pi=5/12pi rad#

and

#5/12pi=3/4pi-pi/3#

Therefore,

#sin(75^@)=sin(5/12pi)#

#=sin(3/4pi-1/3pi)#

#=sin(3/4pi)cos(1/3pi)-cos(3/4pi)sin(1/3pi)#

#=sqrt2/2*1/2+sqrt2/2*sqrt3/2#

#=(sqrt6+sqrt2)/4#

#cos(75^@)=cos(5/12pi)=cos(3/4pi-1/3pi)#

#=cos(3/4pi)cos(1/3pi)+sin(3/4pi)sin(1/3pi)#

#=-sqrt2/2*1/2+sqrt2/2*sqrt3/2#

#=(sqrt6-sqrt2)/4#

Therefore,

#tan(75^@)=sin(75^@)/cos(75^@)=(sqrt6+sqrt2)/(sqrt6-sqrt2)#