解答
12⋅9.8⋅sin(a)−0.6⋅12⋅9.8⋅cos(a)=12⋅1.79
解答
a=−2.75844…+2πn,a=0.69769…+2πn
+1
度数
a=−158.04722…∘+360∘n,a=39.97473…∘+360∘n求解步骤
12⋅9.8sin(a)−0.6⋅12⋅9.8cos(a)=12⋅1.79
两边加上 0.6129.8cos(a)117.6sin(a)=21.48+70.56cos(a)
两边进行平方(117.6sin(a))2=(21.48+70.56cos(a))2
两边减去 (21.48+70.56cos(a))213829.76sin2(a)−461.3904−3031.2576cos(a)−4978.7136cos2(a)=0
使用三角恒等式改写
−461.3904+13829.76sin2(a)−3031.2576cos(a)−4978.7136cos2(a)
使用毕达哥拉斯恒等式: cos2(x)+sin2(x)=1sin2(x)=1−cos2(x)=−461.3904+13829.76(1−cos2(a))−3031.2576cos(a)−4978.7136cos2(a)
化简 −461.3904+13829.76(1−cos2(a))−3031.2576cos(a)−4978.7136cos2(a):−18808.4736cos2(a)−3031.2576cos(a)+13368.3696
−461.3904+13829.76(1−cos2(a))−3031.2576cos(a)−4978.7136cos2(a)
乘开 13829.76(1−cos2(a)):13829.76−13829.76cos2(a)
13829.76(1−cos2(a))
使用分配律: a(b−c)=ab−aca=13829.76,b=1,c=cos2(a)=13829.76⋅1−13829.76cos2(a)
=1⋅13829.76−13829.76cos2(a)
数字相乘:1⋅13829.76=13829.76=13829.76−13829.76cos2(a)
=−461.3904+13829.76−13829.76cos2(a)−3031.2576cos(a)−4978.7136cos2(a)
化简 −461.3904+13829.76−13829.76cos2(a)−3031.2576cos(a)−4978.7136cos2(a):−18808.4736cos2(a)−3031.2576cos(a)+13368.3696
−461.3904+13829.76−13829.76cos2(a)−3031.2576cos(a)−4978.7136cos2(a)
对同类项分组=−13829.76cos2(a)−3031.2576cos(a)−4978.7136cos2(a)−461.3904+13829.76
同类项相加:−13829.76cos2(a)−4978.7136cos2(a)=−18808.4736cos2(a)=−18808.4736cos2(a)−3031.2576cos(a)−461.3904+13829.76
数字相加/相减:−461.3904+13829.76=13368.3696=−18808.4736cos2(a)−3031.2576cos(a)+13368.3696
=−18808.4736cos2(a)−3031.2576cos(a)+13368.3696
=−18808.4736cos2(a)−3031.2576cos(a)+13368.3696
13368.3696−18808.4736cos2(a)−3031.2576cos(a)=0
用替代法求解
13368.3696−18808.4736cos2(a)−3031.2576cos(a)=0
令:cos(a)=u13368.3696−18808.4736u2−3031.2576u=0
13368.3696−18808.4736u2−3031.2576u=0:u=−37616.94723031.2576+1014943029.424128,u=37616.94721014943029.424128−3031.2576
13368.3696−18808.4736u2−3031.2576u=0
改写成标准形式 ax2+bx+c=0−18808.4736u2−3031.2576u+13368.3696=0
使用求根公式求解
−18808.4736u2−3031.2576u+13368.3696=0
二次方程求根公式:
若 a=−18808.4736,b=−3031.2576,c=13368.3696u1,2=2(−18808.4736)−(−3031.2576)±(−3031.2576)2−4(−18808.4736)⋅13368.3696
u1,2=2(−18808.4736)−(−3031.2576)±(−3031.2576)2−4(−18808.4736)⋅13368.3696
(−3031.2576)2−4(−18808.4736)⋅13368.3696=1014943029.424128
(−3031.2576)2−4(−18808.4736)⋅13368.3696
使用法则 −(−a)=a=(−3031.2576)2+4⋅18808.4736⋅13368.3696
使用指数法则: (−a)n=an,若 n 是偶数(−3031.2576)2=3031.25762=3031.25762+4⋅13368.3696⋅18808.4736
数字相乘:4⋅18808.4736⋅13368.3696=1005754506.78657…=3031.25762+1005754506.78657…
3031.25762=9188522.63755…=9188522.63755…+1005754506.78657…
数字相加:9188522.63755…+1005754506.78657…=1014943029.424128=1014943029.424128
u1,2=2(−18808.4736)−(−3031.2576)±1014943029.424128
将解分隔开u1=2(−18808.4736)−(−3031.2576)+1014943029.424128,u2=2(−18808.4736)−(−3031.2576)−1014943029.424128
u=2(−18808.4736)−(−3031.2576)+1014943029.424128:−37616.94723031.2576+1014943029.424128
2(−18808.4736)−(−3031.2576)+1014943029.424128
去除括号: (−a)=−a,−(−a)=a=−2⋅18808.47363031.2576+1014943029.424128
数字相乘:2⋅18808.4736=37616.9472=−37616.94723031.2576+1014943029.424128
使用分式法则: −ba=−ba=−37616.94723031.2576+1014943029.424128
u=2(−18808.4736)−(−3031.2576)−1014943029.424128:37616.94721014943029.424128−3031.2576
2(−18808.4736)−(−3031.2576)−1014943029.424128
去除括号: (−a)=−a,−(−a)=a=−2⋅18808.47363031.2576−1014943029.424128
数字相乘:2⋅18808.4736=37616.9472=−37616.94723031.2576−1014943029.424128
使用分式法则: −b−a=ba3031.2576−1014943029.424128=−(1014943029.424128−3031.2576)=37616.94721014943029.424128−3031.2576
二次方程组的解是:u=−37616.94723031.2576+1014943029.424128,u=37616.94721014943029.424128−3031.2576
u=cos(a)代回cos(a)=−37616.94723031.2576+1014943029.424128,cos(a)=37616.94721014943029.424128−3031.2576
cos(a)=−37616.94723031.2576+1014943029.424128,cos(a)=37616.94721014943029.424128−3031.2576
cos(a)=−37616.94723031.2576+1014943029.424128:a=arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=−arccos(−37616.94723031.2576+1014943029.424128)+2πn
cos(a)=−37616.94723031.2576+1014943029.424128
使用反三角函数性质
cos(a)=−37616.94723031.2576+1014943029.424128
cos(a)=−37616.94723031.2576+1014943029.424128的通解cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πna=arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=−arccos(−37616.94723031.2576+1014943029.424128)+2πn
a=arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=−arccos(−37616.94723031.2576+1014943029.424128)+2πn
cos(a)=37616.94721014943029.424128−3031.2576:a=arccos(37616.94721014943029.424128−3031.2576)+2πn,a=2π−arccos(37616.94721014943029.424128−3031.2576)+2πn
cos(a)=37616.94721014943029.424128−3031.2576
使用反三角函数性质
cos(a)=37616.94721014943029.424128−3031.2576
cos(a)=37616.94721014943029.424128−3031.2576的通解cos(x)=a⇒x=arccos(a)+2πn,x=2π−arccos(a)+2πna=arccos(37616.94721014943029.424128−3031.2576)+2πn,a=2π−arccos(37616.94721014943029.424128−3031.2576)+2πn
a=arccos(37616.94721014943029.424128−3031.2576)+2πn,a=2π−arccos(37616.94721014943029.424128−3031.2576)+2πn
合并所有解a=arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=−arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=arccos(37616.94721014943029.424128−3031.2576)+2πn,a=2π−arccos(37616.94721014943029.424128−3031.2576)+2πn
将解代入原方程进行验证
将它们代入 129.8sin(a)−0.6129.8cos(a)=121.79检验解是否符合
去除与方程不符的解。
检验 arccos(−37616.94723031.2576+1014943029.424128)+2πn的解:假
arccos(−37616.94723031.2576+1014943029.424128)+2πn
代入 n=1arccos(−37616.94723031.2576+1014943029.424128)+2π1
对于 129.8sin(a)−0.6129.8cos(a)=121.79代入a=arccos(−37616.94723031.2576+1014943029.424128)+2π112⋅9.8sin(arccos(−37616.94723031.2576+1014943029.424128)+2π1)−0.6⋅12⋅9.8cos(arccos(−37616.94723031.2576+1014943029.424128)+2π1)=12⋅1.79
整理后得109.40771…=21.48
⇒假
检验 −arccos(−37616.94723031.2576+1014943029.424128)+2πn的解:真
−arccos(−37616.94723031.2576+1014943029.424128)+2πn
代入 n=1−arccos(−37616.94723031.2576+1014943029.424128)+2π1
对于 129.8sin(a)−0.6129.8cos(a)=121.79代入a=−arccos(−37616.94723031.2576+1014943029.424128)+2π112⋅9.8sin(−arccos(−37616.94723031.2576+1014943029.424128)+2π1)−0.6⋅12⋅9.8cos(−arccos(−37616.94723031.2576+1014943029.424128)+2π1)=12⋅1.79
整理后得21.48=21.48
⇒真
检验 arccos(37616.94721014943029.424128−3031.2576)+2πn的解:真
arccos(37616.94721014943029.424128−3031.2576)+2πn
代入 n=1arccos(37616.94721014943029.424128−3031.2576)+2π1
对于 129.8sin(a)−0.6129.8cos(a)=121.79代入a=arccos(37616.94721014943029.424128−3031.2576)+2π112⋅9.8sin(arccos(37616.94721014943029.424128−3031.2576)+2π1)−0.6⋅12⋅9.8cos(arccos(37616.94721014943029.424128−3031.2576)+2π1)=12⋅1.79
整理后得21.48=21.48
⇒真
检验 2π−arccos(37616.94721014943029.424128−3031.2576)+2πn的解:假
2π−arccos(37616.94721014943029.424128−3031.2576)+2πn
代入 n=12π−arccos(37616.94721014943029.424128−3031.2576)+2π1
对于 129.8sin(a)−0.6129.8cos(a)=121.79代入a=2π−arccos(37616.94721014943029.424128−3031.2576)+2π112⋅9.8sin(2π−arccos(37616.94721014943029.424128−3031.2576)+2π1)−0.6⋅12⋅9.8cos(2π−arccos(37616.94721014943029.424128−3031.2576)+2π1)=12⋅1.79
整理后得−129.62418…=21.48
⇒假
a=−arccos(−37616.94723031.2576+1014943029.424128)+2πn,a=arccos(37616.94721014943029.424128−3031.2576)+2πn
以小数形式表示解a=−2.75844…+2πn,a=0.69769…+2πn