解答
sin(2x)cos(6x)−cos(2x)sin(6x)=−0.55
解答
x=40.58236…−2πn,x=−4π−40.58236…−2πn
+1
度数
x=8.34175…∘−90∘n,x=−53.34175…∘−90∘n求解步骤
sin(2x)cos(6x)−cos(2x)sin(6x)=−0.55
使用三角恒等式改写
sin(2x)cos(6x)−cos(2x)sin(6x)
使用角差恒等式: sin(s)cos(t)−cos(s)sin(t)=sin(s−t)=sin(2x−6x)
sin(2x−6x)=−0.55
使用反三角函数性质
sin(2x−6x)=−0.55
sin(2x−6x)=−0.55的通解sin(x)=−a⇒x=arcsin(−a)+2πn,x=π+arcsin(a)+2πn2x−6x=arcsin(−0.55)+2πn,2x−6x=π+arcsin(0.55)+2πn
2x−6x=arcsin(−0.55)+2πn,2x−6x=π+arcsin(0.55)+2πn
解 2x−6x=arcsin(−0.55)+2πn:x=4arcsin(2011)−2πn
2x−6x=arcsin(−0.55)+2πn
化简 2x−6x:−4x
2x−6x
同类项相加:2x−6x=−4x=−4x
化简 arcsin(−0.55)+2πn:−arcsin(2011)+2πn
arcsin(−0.55)+2πn
arcsin(−0.55)=−arcsin(2011)
arcsin(−0.55)
=arcsin(−2011)
利用以下特性:arcsin(−x)=−arcsin(x)arcsin(−2011)=−arcsin(2011)=−arcsin(2011)
=−arcsin(2011)+2πn
−4x=−arcsin(2011)+2πn
两边除以 −4
−4x=−arcsin(2011)+2πn
两边除以 −4−4−4x=−−4arcsin(2011)+−42πn
化简
−4−4x=−−4arcsin(2011)+−42πn
化简 −4−4x:x
−4−4x
使用分式法则: −b−a=ba=44x
数字相除:44=1=x
化简 −−4arcsin(2011)+−42πn:4arcsin(2011)−2πn
−−4arcsin(2011)+−42πn
−4arcsin(2011)=−4arcsin(2011)
−4arcsin(2011)
使用分式法则: −ba=−ba=−4arcsin(2011)
−42πn=−2πn
−42πn
使用分式法则: −ba=−ba=−42πn
约分:2=−2πn
=−(−4arcsin(2011))−2πn
使用法则 −(−a)=a=4arcsin(2011)−2πn
x=4arcsin(2011)−2πn
x=4arcsin(2011)−2πn
x=4arcsin(2011)−2πn
解 2x−6x=π+arcsin(0.55)+2πn:x=−4π−4arcsin(0.55)−2πn
2x−6x=π+arcsin(0.55)+2πn
同类项相加:2x−6x=−4x−4x=π+arcsin(0.55)+2πn
两边除以 −4
−4x=π+arcsin(0.55)+2πn
两边除以 −4−4−4x=−4π+−4arcsin(0.55)+−42πn
化简
−4−4x=−4π+−4arcsin(0.55)+−42πn
化简 −4−4x:x
−4−4x
使用分式法则: −b−a=ba=44x
数字相除:44=1=x
化简 −4π+−4arcsin(0.55)+−42πn:−4π−4arcsin(0.55)−2πn
−4π+−4arcsin(0.55)+−42πn
使用分式法则: −ba=−ba=−4π+−4arcsin(0.55)+−42πn
使用分式法则: −ba=−ba=−4π−4arcsin(0.55)+−42πn
−42πn=−2πn
−42πn
使用分式法则: −ba=−ba=−42πn
约分:2=−2πn
=−4π−4arcsin(0.55)−2πn
x=−4π−4arcsin(0.55)−2πn
x=−4π−4arcsin(0.55)−2πn
x=−4π−4arcsin(0.55)−2πn
x=4arcsin(2011)−2πn,x=−4π−4arcsin(0.55)−2πn
以小数形式表示解x=40.58236…−2πn,x=−4π−40.58236…−2πn