解答
tan2(x)=5sin2(x)
解答
x=2πn,x=π+2πn,x=2.03444…+2πn,x=−2.03444…+2πn,x=1.10714…+2πn,x=2π−1.10714…+2πn
+1
度数
x=0∘+360∘n,x=180∘+360∘n,x=116.56505…∘+360∘n,x=−116.56505…∘+360∘n,x=63.43494…∘+360∘n,x=296.56505…∘+360∘n求解步骤
tan2(x)=5sin2(x)
两边减去 5sin2(x)tan2(x)−5sin2(x)=0
分解 tan2(x)−5sin2(x):(tan(x)+5sin(x))(tan(x)−5sin(x))
tan2(x)−5sin2(x)
将 tan2(x)−5sin2(x) 改写为 tan2(x)−(5sin(x))2
tan2(x)−5sin2(x)
使用根式运算法则: a=(a)25=(5)2=tan2(x)−(5)2sin2(x)
使用指数法则: ambm=(ab)m(5)2sin2(x)=(5sin(x))2=tan2(x)−(5sin(x))2
=tan2(x)−(5sin(x))2
使用平方差公式: x2−y2=(x+y)(x−y)tan2(x)−(5sin(x))2=(tan(x)+5sin(x))(tan(x)−5sin(x))=(tan(x)+5sin(x))(tan(x)−5sin(x))
(tan(x)+5sin(x))(tan(x)−5sin(x))=0
分别求解每个部分tan(x)+5sin(x)=0ortan(x)−5sin(x)=0
tan(x)+5sin(x)=0:x=2πn,x=π+2πn,x=arccos(−55)+2πn,x=−arccos(−55)+2πn
tan(x)+5sin(x)=0
用 sin, cos 表示
tan(x)+sin(x)5
使用基本三角恒等式: tan(x)=cos(x)sin(x)=cos(x)sin(x)+sin(x)5
化简 cos(x)sin(x)+sin(x)5:cos(x)sin(x)+5sin(x)cos(x)
cos(x)sin(x)+sin(x)5
将项转换为分式: 5sin(x)=cos(x)sin(x)5cos(x)=cos(x)sin(x)+cos(x)sin(x)5cos(x)
因为分母相等，所以合并分式: ca±cb=ca±b=cos(x)sin(x)+sin(x)5cos(x)
=cos(x)sin(x)+5sin(x)cos(x)
cos(x)sin(x)+cos(x)sin(x)5=0
g(x)f(x)=0⇒f(x)=0sin(x)+cos(x)sin(x)5=0
分解 sin(x)+cos(x)sin(x)5:sin(x)(1+5cos(x))
sin(x)+cos(x)sin(x)5
因式分解出通项 sin(x)=sin(x)(1+cos(x)5)
sin(x)(1+5cos(x))=0
分别求解每个部分sin(x)=0or1+5cos(x)=0
sin(x)=0:x=2πn,x=π+2πn
sin(x)=0
sin(x)=0的通解
sin(x) 周期表（周期为 2πn"）：
x06π4π3π2π32π43π65πsin(x)02122231232221xπ67π45π34π23π35π47π611πsin(x)0−21−22−23−1−23−22−21
x=0+2πn,x=π+2πn
x=0+2πn,x=π+2πn
解 x=0+2πn:x=2πn
x=0+2πn
0+2πn=2πnx=2πn
x=2πn,x=π+2πn
1+5cos(x)=0:x=arccos(−55)+2πn,x=−arccos(−55)+2πn
1+5cos(x)=0
将 1到右边
1+5cos(x)=0
两边减去 11+5cos(x)−1=0−1
化简5cos(x)=−1
5cos(x)=−1
两边除以 5
5cos(x)=−1
两边除以 555cos(x)=5−1
化简
55cos(x)=5−1
化简 55cos(x):cos(x)
55cos(x)
约分:5=cos(x)
化简 5−1:−55
5−1
使用分式法则: b−a=−ba=−51
−51有理化:−55
−51
乘以共轭根式 55=−551⋅5
1⋅5=5
55=5
55
使用根式运算法则: aa=a55=5=5
=−55
=−55
cos(x)=−55
cos(x)=−55
cos(x)=−55
使用反三角函数性质
cos(x)=−55
cos(x)=−55的通解cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πnx=arccos(−55)+2πn,x=−arccos(−55)+2πn
x=arccos(−55)+2πn,x=−arccos(−55)+2πn
合并所有解x=2πn,x=π+2πn,x=arccos(−55)+2πn,x=−arccos(−55)+2πn
tan(x)−5sin(x)=0:x=2πn,x=π+2πn,x=arccos(55)+2πn,x=2π−arccos(55)+2πn
tan(x)−5sin(x)=0
用 sin, cos 表示
tan(x)−sin(x)5
使用基本三角恒等式: tan(x)=cos(x)sin(x)=cos(x)sin(x)−sin(x)5
化简 cos(x)sin(x)−sin(x)5:cos(x)sin(x)−5sin(x)cos(x)
cos(x)sin(x)−sin(x)5
将项转换为分式: 5sin(x)=cos(x)sin(x)5cos(x)=cos(x)sin(x)−cos(x)sin(x)5cos(x)
因为分母相等，所以合并分式: ca±cb=ca±b=cos(x)sin(x)−sin(x)5cos(x)
=cos(x)sin(x)−5sin(x)cos(x)
cos(x)sin(x)−cos(x)sin(x)5=0
g(x)f(x)=0⇒f(x)=0sin(x)−cos(x)sin(x)5=0
分解 sin(x)−cos(x)sin(x)5:sin(x)(1−5cos(x))
sin(x)−cos(x)sin(x)5
因式分解出通项 sin(x)=sin(x)(1−cos(x)5)
sin(x)(1−5cos(x))=0
分别求解每个部分sin(x)=0or1−5cos(x)=0
sin(x)=0:x=2πn,x=π+2πn
sin(x)=0
sin(x)=0的通解
sin(x) 周期表（周期为 2πn"）：
x06π4π3π2π32π43π65πsin(x)02122231232221xπ67π45π34π23π35π47π611πsin(x)0−21−22−23−1−23−22−21
x=0+2πn,x=π+2πn
x=0+2πn,x=π+2πn
解 x=0+2πn:x=2πn
x=0+2πn
0+2πn=2πnx=2πn
x=2πn,x=π+2πn
1−5cos(x)=0:x=arccos(55)+2πn,x=2π−arccos(55)+2πn
1−5cos(x)=0
将 1到右边
1−5cos(x)=0
两边减去 11−5cos(x)−1=0−1
化简−5cos(x)=−1
−5cos(x)=−1
两边除以 −5
−5cos(x)=−1
两边除以 −5−5−5cos(x)=−5−1
化简
−5−5cos(x)=−5−1
化简 −5−5cos(x):cos(x)
−5−5cos(x)
使用分式法则: −b−a=ba=55cos(x)
约分:5=cos(x)
化简 −5−1:55
−5−1
使用分式法则: −b−a=ba=51
51有理化:55
51
乘以共轭根式 55=551⋅5
1⋅5=5
55=5
55
使用根式运算法则: aa=a55=5=5
=55
=55
cos(x)=55
cos(x)=55
cos(x)=55
使用反三角函数性质
cos(x)=55
cos(x)=55的通解cos(x)=a⇒x=arccos(a)+2πn,x=2π−arccos(a)+2πnx=arccos(55)+2πn,x=2π−arccos(55)+2πn
x=arccos(55)+2πn,x=2π−arccos(55)+2πn
合并所有解x=2πn,x=π+2πn,x=arccos(55)+2πn,x=2π−arccos(55)+2πn
合并所有解x=2πn,x=π+2πn,x=arccos(−55)+2πn,x=−arccos(−55)+2πn,x=arccos(55)+2πn,x=2π−arccos(55)+2πn
以小数形式表示解x=2πn,x=π+2πn,x=2.03444…+2πn,x=−2.03444…+2πn,x=1.10714…+2πn,x=2π−1.10714…+2πn