1) Use half-angle identity for sine:
sin2a=±21−cosa Find cosa:
cosa=±1−sin2acosa=±1−(54)2=±1−2516=±259=±53Since a is in the quadrant I, cosa>0,cosa=53
Hence,
sin2a=±21−53=±252=±51 Since a is in the quadrant I, sin2a>0,sin2a=51
2) Use half-angle identity for cosine:
cos2a=±21+cosacos2a=±21+53=±258=±54
Since a is in the quadrant I, cos2a>0,cos2a=54
3) Half-angle for tangent and cotangent:
tan2a=sin2a:cos2a=51:54=41=21cot2a=cos2a:sin2a=54:51=14=2
4) Use double-angle identity for sine:
sin2a=2sinacosasin2a=2⋅54⋅53=25245) Use double-angle identity for cosine:
cos2a=1−2sin2a=1−2⋅(54)2=1−2532=−257 6) Double-angle for tangent and cotangent:
tan2a=sin2a:cos2a=2524:(−257)=−724
cot2a=cos2a:sin2a=−257:2524=−247
Answer: sin2a=51,cos2a=54,tan2a=21,cot2a=2,
sin2a=2524,cos2a=−257,tan2a=−724,cot2a=−247
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