解答
cos2(111∘)+cos2(69.3∘)+cos2(x)=1
解答
x=0.52748…+360∘n,x=360∘−0.52748…+360∘n,x=2.61410…+360∘n,x=−2.61410…+360∘n
+1
弧度
x=0.52748…+2πn,x=2π−0.52748…+2πn,x=2.61410…+2πn,x=−2.61410…+2πn求解步骤
cos2(111∘)+cos2(69.3∘)+cos2(x)=1
用替代法求解
cos2(111∘)+cos2(69.3∘)+cos2(x)=1
令:cos(x)=ucos2(111∘)+cos2(69.3∘)+u2=1
cos2(111∘)+cos2(69.3∘)+u2=1:u=1−cos2(111∘)−cos2(69.3∘),u=−1−cos2(111∘)−cos2(69.3∘)
cos2(111∘)+cos2(69.3∘)+u2=1
将 cos2(111∘)到右边
cos2(111∘)+cos2(69.3∘)+u2=1
两边减去 cos2(111∘)cos2(111∘)+cos2(69.3∘)+u2−cos2(111∘)=1−cos2(111∘)
化简cos2(69.3∘)+u2=1−cos2(111∘)
cos2(69.3∘)+u2=1−cos2(111∘)
将 cos2(69.3∘)到右边
cos2(69.3∘)+u2=1−cos2(111∘)
两边减去 cos2(69.3∘)cos2(69.3∘)+u2−cos2(69.3∘)=1−cos2(111∘)−cos2(69.3∘)
化简u2=1−cos2(111∘)−cos2(69.3∘)
u2=1−cos2(111∘)−cos2(69.3∘)
对于 x2=f(a) 解为 x=f(a),−f(a)
u=1−cos2(111∘)−cos2(69.3∘),u=−1−cos2(111∘)−cos2(69.3∘)
u=cos(x)代回cos(x)=1−cos2(111∘)−cos2(69.3∘),cos(x)=−1−cos2(111∘)−cos2(69.3∘)
cos(x)=1−cos2(111∘)−cos2(69.3∘),cos(x)=−1−cos2(111∘)−cos2(69.3∘)
cos(x)=1−cos2(111∘)−cos2(69.3∘):x=arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n,x=360∘−arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n
cos(x)=1−cos2(111∘)−cos2(69.3∘)
使用反三角函数性质
cos(x)=1−cos2(111∘)−cos2(69.3∘)
cos(x)=1−cos2(111∘)−cos2(69.3∘)的通解cos(x)=a⇒x=arccos(a)+360∘n,x=360∘−arccos(a)+360∘nx=arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n,x=360∘−arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n
x=arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n,x=360∘−arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n
cos(x)=−1−cos2(111∘)−cos2(69.3∘):x=arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n,x=−arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n
cos(x)=−1−cos2(111∘)−cos2(69.3∘)
使用反三角函数性质
cos(x)=−1−cos2(111∘)−cos2(69.3∘)
cos(x)=−1−cos2(111∘)−cos2(69.3∘)的通解cos(x)=−a⇒x=arccos(−a)+360∘n,x=−arccos(−a)+360∘nx=arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n,x=−arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n
x=arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n,x=−arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n
合并所有解x=arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n,x=360∘−arccos(1−cos2(111∘)−cos2(69.3∘))+360∘n,x=arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n,x=−arccos(−1−cos2(111∘)−cos2(69.3∘))+360∘n
以小数形式表示解x=0.52748…+360∘n,x=360∘−0.52748…+360∘n,x=2.61410…+360∘n,x=−2.61410…+360∘n