三角恒等变换1函数y=sin ^4+cos ^2的最小正周期是2已知在三角形ABC中,3sinA+4cosB=6,4si

1、先化简函数,再根据公式即可求出周期：
y
=(sin²x)²+cos²x
=[(1-cos2x)/2]²+(1+cos2x)/2
=(3+cos²2x)/4
=[3+(1+cos4x)/2]/4
=(1/8)*(7+cos4x)
∴周期T=2π/4=π/2,
2、两式平方后相加即可：
(3sinA+4cosB)²+(4sinB+3cosA)²
=(9sin²A+9cos²A)+(16cos²B+16sin²B)+(24sinAcosB+24sinBcosA)
=9+16+24sin(A+B)
=25+24sin[π-(A+B)]
=25+24sinC
=6²+1²
=37
∴sinC=1/2
∴∠C=π/6或5π/6
3、(sin65°+sin15°sin10°)／(sin25°-cos15°cos80°)
=(cos25°+sin15°cos10°)/[sin(15°+10°)-cos15°sin10°]
=[cos(15°+10°)+sin15°cos10°]/[(sin15°cos10°+sin10°cos15°)-cos15°sin10°]
=[(cos15°cos10°-sin15°sin10°)+sin15°sin10°]/(sin15°cos10°)
=(cos15°cos10°)/(sin15°cos10°)
=cot15°
4、乘以cos6°再计算：
sin6°sin42°sin66°sin78°
=cos6°sin6°sin42°sin66°sin78°/cos6°
=(1/2)sin12°cos12°cos24°cos48°/cos6°
=(1/4)sin24°cos24°cos48°/cos6°
=(1/8)sin48°cos48°/cos6°
=(1/16)sin96°/cos6°
=(1/16)sin84°/sin84°
=1/16
5、sin²20°+cos²50°+sin20°cos50°
=(1-cos40°)/2+(1+cos100°)/2+(1/2)*(sin70°-sin30°)
=(1-cos40°)/2+(1-cos80°)/2+(1/2)*(cos20°-1/2)
=1-(1/2)*(cos40°+cos80°)+(1/2)*(cos20°-1/2)
=1-(cos60°cos20°)+(1/2)cos20°-1/4
=1-(1/2)cos20°+(1/2)cos20°-1/4
=3/4
6、作和后,真数部分乘以sin(π/9)再算:
log2 cosπ/9+ log2 cos2π/9+ log2 cos4π/9
=log2 [(cosπ/9)*(cos2π/9)*(cos4π/9)]
=log2 [(sinπ/9)*(cosπ/9)*(cos2π/9)*(cos4π/9)/(sinπ/9)]
=log2 [(1/8)*sin(8π/9)/(sinπ/9)]
=log2 (1/8)
=-3
7、(1+tanA)(1+tanB)
=1+tanA+tanB+tanAtanB
=1+tan(A+B)(1-tanAtanB)+tanAtanB
=2