解答
21.16=19.6sin(x)−29.4cos(x)
解答
x=−2.80086…+2πn,x=1.62485…+2πn
+1
度数
x=−160.47763…∘+360∘n,x=93.09749…∘+360∘n求解步骤
21.16=19.6sin(x)−29.4cos(x)
两边加上 29.4cos(x)19.6sin(x)=21.16+29.4cos(x)
两边进行平方(19.6sin(x))2=(21.16+29.4cos(x))2
两边减去 (21.16+29.4cos(x))2384.16sin2(x)−447.7456−1244.208cos(x)−864.36cos2(x)=0
使用三角恒等式改写
−447.7456−1244.208cos(x)+384.16sin2(x)−864.36cos2(x)
使用毕达哥拉斯恒等式: cos2(x)+sin2(x)=1sin2(x)=1−cos2(x)=−447.7456−1244.208cos(x)+384.16(1−cos2(x))−864.36cos2(x)
化简 −447.7456−1244.208cos(x)+384.16(1−cos2(x))−864.36cos2(x):−1248.52cos2(x)−1244.208cos(x)−63.5856
−447.7456−1244.208cos(x)+384.16(1−cos2(x))−864.36cos2(x)
乘开 384.16(1−cos2(x)):384.16−384.16cos2(x)
384.16(1−cos2(x))
使用分配律: a(b−c)=ab−aca=384.16,b=1,c=cos2(x)=384.16⋅1−384.16cos2(x)
=1⋅384.16−384.16cos2(x)
数字相乘:1⋅384.16=384.16=384.16−384.16cos2(x)
=−447.7456−1244.208cos(x)+384.16−384.16cos2(x)−864.36cos2(x)
化简 −447.7456−1244.208cos(x)+384.16−384.16cos2(x)−864.36cos2(x):−1248.52cos2(x)−1244.208cos(x)−63.5856
−447.7456−1244.208cos(x)+384.16−384.16cos2(x)−864.36cos2(x)
同类项相加:−384.16cos2(x)−864.36cos2(x)=−1248.52cos2(x)=−447.7456−1244.208cos(x)+384.16−1248.52cos2(x)
对同类项分组=−1244.208cos(x)−1248.52cos2(x)−447.7456+384.16
数字相加/相减:−447.7456+384.16=−63.5856=−1248.52cos2(x)−1244.208cos(x)−63.5856
=−1248.52cos2(x)−1244.208cos(x)−63.5856
=−1248.52cos2(x)−1244.208cos(x)−63.5856
−63.5856−1244.208cos(x)−1248.52cos2(x)=0
用替代法求解
−63.5856−1244.208cos(x)−1248.52cos2(x)=0
令:cos(x)=u−63.5856−1244.208u−1248.52u2=0
−63.5856−1244.208u−1248.52u2=0:u=−2497.041244.208+1230501.974016,u=−2497.041244.208−1230501.974016
−63.5856−1244.208u−1248.52u2=0
改写成标准形式 ax2+bx+c=0−1248.52u2−1244.208u−63.5856=0
使用求根公式求解
−1248.52u2−1244.208u−63.5856=0
二次方程求根公式:
若 a=−1248.52,b=−1244.208,c=−63.5856u1,2=2(−1248.52)−(−1244.208)±(−1244.208)2−4(−1248.52)(−63.5856)
u1,2=2(−1248.52)−(−1244.208)±(−1244.208)2−4(−1248.52)(−63.5856)
(−1244.208)2−4(−1248.52)(−63.5856)=1230501.974016
(−1244.208)2−4(−1248.52)(−63.5856)
使用法则 −(−a)=a=(−1244.208)2−4⋅1248.52⋅63.5856
使用指数法则: (−a)n=an,若 n 是偶数(−1244.208)2=1244.2082=1244.2082−4⋅63.5856⋅1248.52
数字相乘:4⋅1248.52⋅63.5856=317551.573248=1244.2082−317551.573248
1244.2082=1548053.547264=1548053.547264−317551.573248
数字相减:1548053.547264−317551.573248=1230501.974016=1230501.974016
u1,2=2(−1248.52)−(−1244.208)±1230501.974016
将解分隔开u1=2(−1248.52)−(−1244.208)+1230501.974016,u2=2(−1248.52)−(−1244.208)−1230501.974016
u=2(−1248.52)−(−1244.208)+1230501.974016:−2497.041244.208+1230501.974016
2(−1248.52)−(−1244.208)+1230501.974016
去除括号: (−a)=−a,−(−a)=a=−2⋅1248.521244.208+1230501.974016
数字相乘:2⋅1248.52=2497.04=−2497.041244.208+1230501.974016
使用分式法则: −ba=−ba=−2497.041244.208+1230501.974016
u=2(−1248.52)−(−1244.208)−1230501.974016:−2497.041244.208−1230501.974016
2(−1248.52)−(−1244.208)−1230501.974016
去除括号: (−a)=−a,−(−a)=a=−2⋅1248.521244.208−1230501.974016
数字相乘:2⋅1248.52=2497.04=−2497.041244.208−1230501.974016
使用分式法则: −ba=−ba=−2497.041244.208−1230501.974016
二次方程组的解是:u=−2497.041244.208+1230501.974016,u=−2497.041244.208−1230501.974016
u=cos(x)代回cos(x)=−2497.041244.208+1230501.974016,cos(x)=−2497.041244.208−1230501.974016
cos(x)=−2497.041244.208+1230501.974016,cos(x)=−2497.041244.208−1230501.974016
cos(x)=−2497.041244.208+1230501.974016:x=arccos(−2497.041244.208+1230501.974016)+2πn,x=−arccos(−2497.041244.208+1230501.974016)+2πn
cos(x)=−2497.041244.208+1230501.974016
使用反三角函数性质
cos(x)=−2497.041244.208+1230501.974016
cos(x)=−2497.041244.208+1230501.974016的通解cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πnx=arccos(−2497.041244.208+1230501.974016)+2πn,x=−arccos(−2497.041244.208+1230501.974016)+2πn
x=arccos(−2497.041244.208+1230501.974016)+2πn,x=−arccos(−2497.041244.208+1230501.974016)+2πn
cos(x)=−2497.041244.208−1230501.974016:x=arccos(−2497.041244.208−1230501.974016)+2πn,x=−arccos(−2497.041244.208−1230501.974016)+2πn
cos(x)=−2497.041244.208−1230501.974016
使用反三角函数性质
cos(x)=−2497.041244.208−1230501.974016
cos(x)=−2497.041244.208−1230501.974016的通解cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πnx=arccos(−2497.041244.208−1230501.974016)+2πn,x=−arccos(−2497.041244.208−1230501.974016)+2πn
x=arccos(−2497.041244.208−1230501.974016)+2πn,x=−arccos(−2497.041244.208−1230501.974016)+2πn
合并所有解x=arccos(−2497.041244.208+1230501.974016)+2πn,x=−arccos(−2497.041244.208+1230501.974016)+2πn,x=arccos(−2497.041244.208−1230501.974016)+2πn,x=−arccos(−2497.041244.208−1230501.974016)+2πn
将解代入原方程进行验证
将它们代入 19.6sin(x)−29.4cos(x)=21.16检验解是否符合
去除与方程不符的解。
检验 arccos(−2497.041244.208+1230501.974016)+2πn的解:假
arccos(−2497.041244.208+1230501.974016)+2πn
代入 n=1arccos(−2497.041244.208+1230501.974016)+2π1
对于 19.6sin(x)−29.4cos(x)=21.16代入x=arccos(−2497.041244.208+1230501.974016)+2π119.6sin(arccos(−2497.041244.208+1230501.974016)+2π1)−29.4cos(arccos(−2497.041244.208+1230501.974016)+2π1)=21.16
整理后得34.25965…=21.16
⇒假
检验 −arccos(−2497.041244.208+1230501.974016)+2πn的解:真
−arccos(−2497.041244.208+1230501.974016)+2πn
代入 n=1−arccos(−2497.041244.208+1230501.974016)+2π1
对于 19.6sin(x)−29.4cos(x)=21.16代入x=−arccos(−2497.041244.208+1230501.974016)+2π119.6sin(−arccos(−2497.041244.208+1230501.974016)+2π1)−29.4cos(−arccos(−2497.041244.208+1230501.974016)+2π1)=21.16
整理后得21.16=21.16
⇒真
检验 arccos(−2497.041244.208−1230501.974016)+2πn的解:真
arccos(−2497.041244.208−1230501.974016)+2πn
代入 n=1arccos(−2497.041244.208−1230501.974016)+2π1
对于 19.6sin(x)−29.4cos(x)=21.16代入x=arccos(−2497.041244.208−1230501.974016)+2π119.6sin(arccos(−2497.041244.208−1230501.974016)+2π1)−29.4cos(arccos(−2497.041244.208−1230501.974016)+2π1)=21.16
整理后得21.16=21.16
⇒真
检验 −arccos(−2497.041244.208−1230501.974016)+2πn的解:假
−arccos(−2497.041244.208−1230501.974016)+2πn
代入 n=1−arccos(−2497.041244.208−1230501.974016)+2π1
对于 19.6sin(x)−29.4cos(x)=21.16代入x=−arccos(−2497.041244.208−1230501.974016)+2π119.6sin(−arccos(−2497.041244.208−1230501.974016)+2π1)−29.4cos(−arccos(−2497.041244.208−1230501.974016)+2π1)=21.16
整理后得−17.98273…=21.16
⇒假
x=−arccos(−2497.041244.208+1230501.974016)+2πn,x=arccos(−2497.041244.208−1230501.974016)+2πn
以小数形式表示解x=−2.80086…+2πn,x=1.62485…+2πn