Question 444897: How to find the exact value of the angle 67.5 using half-angle identities? It says for sin, cos, and tan... Thanks!
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! How to find the exact value of the angle 67.5 using half-angle identities? It says for sin, cos, and tan... Thanks!
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Half-Angle formulas
sin s/2=+-sqrt((1-cos s)/2), (note:+ or - answer depends on value of sin s/2 not sin s.)
cos s/2=+-sqrt((1+cos s)/2), (note:+ or - answer depends on value of cos s/2 not cos s.)
tan s/2=sin s/(1+cos s)
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sin 67.5=sin 135/2=sqrt((1-cos 135)/2)
=sqrt((1+√2/2)/2)=sqrt((2+√2)/4)
=sqrt(2+√2)/2
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cos 67.5=cos 135/2=sqrt((1+cos 135)/2
=sqrt((1+(-√2/2))/2)=sqrt((1-√2/2))/2)
=sqrt((2-√2)/4)
=sqrt(2-√2)/2
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tan 67.5=tan 135/2=sin 135/(1+cos 135)
=(√2/2)/1+(-√2/2)
=√2/2)/1-(√2/2))
=√2/(2-√2)
Note: ( + answers selected for sin & cos because 67.5º is in quadrant I where sin & cos are positive)
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