Question

已知tana=2求下列各式的值，sinacosa，2sin平方a-3sinacosa-4cos平方a

Answer 1

sinacosa
=1/2*sin2a
=1/2*2tana/(1+tan²a)
=tana/(1+tan²a)
=2/(1+2²)
=2/5

2sin²a-3sinacosa-4cos²a
=2(1-cos²a)-3sinacosa-4cos²a
=2-2cos²a-3sinacosa-4cos²a
=2-6cos²a-3sinacosa
=2-6(1+cos2a)/2-3/2*sin2a
=2-3(1+cos2a)-3/2*sin2a
=2-3-3cos2a-3/2*sin2a
=-1-3*(1-tan²a)/(1+tan²a)-3/2*2/5
=-1-3*(1-2²)/(1+2²)-3/5
=-1+9/5-3/5
=1/5

Answer 2

∵tana＝2
∴sec²a＝1＋tan²a＝5 ∴cos²a＝1/5 sin²a＝4/5 cos(2a)＝2cos²a－1＝﹣3/5
tan(2a)＝2tana/(1－tan²a)＝﹣4/3 sin(2a)＝tan(2a)cos(2a)＝4/5
∴sinacosa＝1/2sin(2a)＝2/5
2sin²a－3sinacosa－4cos²a＝2sin²a－3/2×sin(2a)－4cos²a
＝2×4/5－3/2×4/5－4×1/5＝﹣2/5