Question 816649: Find the exact value of the sin cos and tan of the angle -225 and 13pie/12
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Find the exact value of the sin cos and tan of the angle -225 and 13pie/12
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reference angle for -225˚ is 45˚ in quadrant II where sin>0,cos<0,tan<0.
tan(-225=tan(45)=-1
sin(-225)=sin(45)=√2/2
cos(-225)=cos(45)=-√2/2
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(13π/12)=(9π/12+4π/12)=(3π/4+π/3)
tan(3π/4)=-1
tan(π/3)=√3
Identity:tan(3π/4+π/3)=(tan(3π/4)+tan(π/3))/(1+tan(3π/4)*tan(π/3)
=(-1+√3)/(1+(-1)*(√3)=(-1+√3)/(1+√3)
tan(13π/12)=2-√3
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sin(3π/4)=√2/2
sin(π/3)=√3/2
cos(3π/4)=-√2/2
cos(π/3)=1/2
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Identity:sin(3π/4+π/3)=sin(3π/4)*cos(π/3)+cos(3π/4)*sin(π/3)
=√2/2*(1/2)+(-√2/2)*√3/2=(√2/4)-√6/4=-√6+√2)/4
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Identity: cos(3π/4+π/3)=cos(3π/4)*cos(π/3)-sin(3π/4)*sin(π/3)
=-√2/2*(1/2)-(√2/2)*√3/2=(-√2/4)-√6/4=-√2-√6)/4=-(√6+√2)/4
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Calculator check:
tan(13π/12)≈0.2679..
exact value as calculated=2-√3≈0.2679.
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sin(13π/12)≈-0.2588..
exact value as calculated=-√6+√2)/4≈-0.2588
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cos(13π/12)≈-0.9659..
exact value as calculated=-(√6+√2)/4≈-0.9659..
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