解答
0.56=tan2(45∘+2x)
解答
x=2⋅0.64243…+360∘n−90∘,x=−2⋅0.64243…+360∘n−90∘
+1
弧度
x=2⋅0.64243…−2π+2πn,x=−2⋅0.64243…−2π+2πn求解步骤
0.56=tan2(45∘+2x)
交换两边tan2(45∘+2x)=0.56
用替代法求解
tan2(45∘+2x)=0.56
令:tan(45∘+2x)=uu2=0.56
u2=0.56:u=0.56,u=−0.56
u2=0.56
对于 x2=f(a) 解为 x=f(a),−f(a)
u=0.56,u=−0.56
u=tan(45∘+2x)代回tan(45∘+2x)=0.56,tan(45∘+2x)=−0.56
tan(45∘+2x)=0.56,tan(45∘+2x)=−0.56
tan(45∘+2x)=0.56:x=2arctan(0.56)+360∘n−90∘
tan(45∘+2x)=0.56
使用反三角函数性质
tan(45∘+2x)=0.56
tan(45∘+2x)=0.56的通解tan(x)=a⇒x=arctan(a)+180∘n45∘+2x=arctan(0.56)+180∘n
45∘+2x=arctan(0.56)+180∘n
解 45∘+2x=arctan(0.56)+180∘n:x=2arctan(0.56)+360∘n−90∘
45∘+2x=arctan(0.56)+180∘n
将 45∘到右边
45∘+2x=arctan(0.56)+180∘n
两边减去 45∘45∘+2x−45∘=arctan(0.56)+180∘n−45∘
化简2x=arctan(0.56)+180∘n−45∘
2x=arctan(0.56)+180∘n−45∘
在两边乘以 2
2x=arctan(0.56)+180∘n−45∘
在两边乘以 222x=2arctan(0.56)+360∘n−2⋅45∘
化简
22x=2arctan(0.56)+360∘n−2⋅45∘
化简 22x:x
22x
数字相除:22=1=x
化简 2arctan(0.56)+360∘n−2⋅45∘:2arctan(0.56)+360∘n−90∘
2arctan(0.56)+360∘n−2⋅45∘
2⋅45∘=90∘
2⋅45∘
分式相乘: a⋅cb=ca⋅b=90∘
约分:2=90∘
=2arctan(0.56)+360∘n−90∘
x=2arctan(0.56)+360∘n−90∘
x=2arctan(0.56)+360∘n−90∘
x=2arctan(0.56)+360∘n−90∘
x=2arctan(0.56)+360∘n−90∘
tan(45∘+2x)=−0.56:x=−2arctan(514)+360∘n−90∘
tan(45∘+2x)=−0.56
使用反三角函数性质
tan(45∘+2x)=−0.56
tan(45∘+2x)=−0.56的通解tan(x)=−a⇒x=arctan(−a)+180∘n45∘+2x=arctan(−0.56)+180∘n
45∘+2x=arctan(−0.56)+180∘n
解 45∘+2x=arctan(−0.56)+180∘n:x=−2arctan(514)+360∘n−90∘
45∘+2x=arctan(−0.56)+180∘n
化简 arctan(−0.56)+180∘n:−arctan(514)+180∘n
arctan(−0.56)+180∘n
arctan(−0.56)=−arctan(514)
arctan(−0.56)
=arctan(−2514)
利用以下特性:arctan(−x)=−arctan(x)arctan(−2514)=−arctan(2514)=−arctan(2514)
=−arctan(514)
=−arctan(514)+180∘n
45∘+2x=−arctan(514)+180∘n
将 45∘到右边
45∘+2x=−arctan(514)+180∘n
两边减去 45∘45∘+2x−45∘=−arctan(514)+180∘n−45∘
化简2x=−arctan(514)+180∘n−45∘
2x=−arctan(514)+180∘n−45∘
在两边乘以 2
2x=−arctan(514)+180∘n−45∘
在两边乘以 222x=−2arctan(514)+360∘n−2⋅45∘
化简
22x=−2arctan(514)+360∘n−2⋅45∘
化简 22x:x
22x
数字相除:22=1=x
化简 −2arctan(514)+360∘n−2⋅45∘:−2arctan(514)+360∘n−90∘
−2arctan(514)+360∘n−2⋅45∘
2⋅45∘=90∘
2⋅45∘
分式相乘: a⋅cb=ca⋅b=90∘
约分:2=90∘
=−2arctan(514)+360∘n−90∘
x=−2arctan(514)+360∘n−90∘
x=−2arctan(514)+360∘n−90∘
x=−2arctan(514)+360∘n−90∘
x=−2arctan(514)+360∘n−90∘
合并所有解x=2arctan(0.56)+360∘n−90∘,x=−2arctan(514)+360∘n−90∘
以小数形式表示解x=2⋅0.64243…+360∘n−90∘,x=−2⋅0.64243…+360∘n−90∘