해법
73500=130000⋅sin(x)+0.15⋅130000⋅cos(x)
해법
x=2.39936…+2πn,x=0.44444…+2πn
+1
도
x=137.47362…∘+360∘n,x=25.46484…∘+360∘n솔루션 단계
73500=130000sin(x)+0.15⋅130000cos(x)
빼다 0.15130000cos(x) 양쪽에서130000sin(x)=73500−19500cos(x)
양쪽을 제곱(130000sin(x))2=(73500−19500cos(x))2
빼다 (73500−19500cos(x))2 양쪽에서1300002sin2(x)−735002+2866500000cos(x)−380250000cos2(x)=0
삼각성을 사용하여 다시 쓰기
−735002+1300002sin2(x)+2866500000cos(x)−380250000cos2(x)
피타고라스 정체성 사용: cos2(x)+sin2(x)=1sin2(x)=1−cos2(x)=−735002+1300002(1−cos2(x))+2866500000cos(x)−380250000cos2(x)
−735002+(1−cos2(x))⋅1300002+2866500000cos(x)−380250000cos2(x)=0
대체로 해결
−735002+(1−cos2(x))⋅1300002+2866500000cos(x)−380250000cos2(x)=0
하게: cos(x)=u−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0
−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0:u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0
−735002+(1−u2)⋅1300002+2866500000u−380250000u2 확장 :−735002+1300002−1300002u2+2866500000u−380250000u2
−735002+(1−u2)⋅1300002+2866500000u−380250000u2
=−735002+1300002(1−u2)+2866500000u−380250000u2
1300002(1−u2)확대한다:1300002−1300002u2
1300002(1−u2)
분배 법칙 적용: a(b−c)=ab−aca=1300002,b=1,c=u2=1300002⋅1−1300002u2
곱하다: 1300002⋅1=1300002=1300002−1300002u2
=−735002+1300002−1300002u2+2866500000u−380250000u2
−735002+1300002−1300002u2+2866500000u−380250000u2=0
표준 양식으로 작성 ax2+bx+c=0−(1300002+380250000)u2+2866500000u−735002+1300002=0
쿼드 공식으로 해결
−(1300002+380250000)u2+2866500000u−735002+1300002=0
4차 방정식 공식:
위해서 a=−1300002−380250000,b=2866500000,c=−735002+1300002u1,2=2(−1300002−380250000)−2866500000±28665000002−4(−1300002−380250000)(−735002+1300002)
u1,2=2(−1300002−380250000)−2866500000±28665000002−4(−1300002−380250000)(−735002+1300002)
28665000002−4(−1300002−380250000)(−735002+1300002)=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
28665000002−4(−1300002−380250000)(−735002+1300002)
28665000002−4(−1300002−380250000)(−735002+1300002)확대한다:28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
28665000002−4(−1300002−380250000)(−735002+1300002)
−4(−1300002−380250000)(−735002+1300002)확대한다:−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
(−1300002−380250000)(−735002+1300002)확대한다:514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
(−1300002−380250000)(−735002+1300002)
호일 방법 적용: (a+b)(c+d)=ac+ad+bc+bda=−1300002,b=−380250000,c=−735002,d=1300002=(−1300002)(−735002)+(−1300002)⋅1300002+(−380250000)(−735002)+(−380250000)⋅1300002
마이너스 플러스 규칙 적용(−a)(−b)=ab,+(−a)=−a=1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000
1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000단순화하세요:514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000
1300002⋅735002=514⋅14958268416
1300002⋅735002
요인 정수 130000=24⋅54⋅13=(24⋅54⋅13)2⋅735002
지수 규칙 적용: (ab)c=acbc(24⋅54⋅13)2=(24)2(54)2⋅132=(24)2(54)2⋅132⋅735002
지수 규칙 적용: (ab)c=abc(24)2=24⋅2,(54)2=54⋅2=24⋅2⋅54⋅2⋅132⋅735002
다듬다=28⋅58⋅132⋅735002
요인 정수 73500=53⋅22⋅147=28⋅58⋅132(22⋅53⋅147)2
지수 규칙 적용: (ab)c=acbc(22⋅53⋅147)2=(22)2(53)2⋅1472=28⋅58⋅132(22)2(53)2⋅1472
지수 규칙 적용: (ab)c=abc(22)2=22⋅2,(53)2=53⋅2=28⋅58⋅132⋅22⋅2⋅53⋅2⋅1472
다듬다=28⋅58⋅132⋅24⋅56⋅1472
지수 규칙 적용: ab⋅ac=ab+c28⋅24=28+4=58⋅132⋅28+4⋅56⋅1472
숫자 추가: 8+4=12=58⋅132⋅212⋅56⋅1472
지수 규칙 적용: ab⋅ac=ab+c58⋅56=58+6=132⋅212⋅58+6⋅1472
숫자 추가: 8+6=14=132⋅212⋅514⋅1472
132=169=514⋅212⋅1472⋅169
212=4096=514⋅1472⋅169⋅4096
1472=21609=514⋅169⋅4096⋅21609
숫자를 곱하시오: 169⋅4096⋅21609=14958268416=514⋅14958268416
1300002⋅1300002=1300004
1300002⋅1300002
지수 규칙 적용: ab⋅ac=ab+c1300002⋅1300002=1300002+2=1300002+2
숫자 추가: 2+2=4=1300004
=514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
=514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
=−4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000)
−4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000)확대한다:−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
−4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000)
괄호 배포=(−4)⋅514⋅14958268416+(−4)(−1300004)+(−4)⋅735002⋅380250000+(−4)(−1300002⋅380250000)
마이너스 플러스 규칙 적용+(−a)=−a,(−a)(−b)=ab=−514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
−514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000단순화하세요:−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
−514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
숫자를 곱하시오: 4⋅14958268416=59833073664=−514⋅59833073664+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
숫자를 곱하시오: 4⋅380250000=1521000000=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u1,2=2(−1300002−380250000)−2866500000±28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
솔루션 분리u1=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u2=2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u=2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000:−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
분수 규칙 적용: −b−a=ba−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000=−(1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
분수 규칙 적용: −ba=−ba=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
2차 방정식의 해는 다음과 같다:u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
뒤로 대체 u=cos(x)cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000:x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
트리거 역속성 적용
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
일반 솔루션 cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πnx=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000:x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
트리거 역속성 적용
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
일반 솔루션 cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000cos(x)=a⇒x=arccos(a)+2πn,x=2π−arccos(a)+2πnx=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
모든 솔루션 결합x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
해법을 원래 방정식에 연결하여 검증
솔루션을 에 연결하여 확인합니다 130000sin(x)+0.15130000cos(x)=73500
방정식에 맞지 않는 것은 제거하십시오.
솔루션 확인 arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn:참
arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
n=1끼우다 arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1
130000sin(x)+0.15130000cos(x)=73500 위한 {\ quad}끼우다{\ quad} x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1130000sin(arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)+0.15⋅130000cos(arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)=73500
다듬다73500=73500
⇒참
솔루션 확인 −arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn:거짓
−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
n=1끼우다 −arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1
130000sin(x)+0.15130000cos(x)=73500 위한 {\ quad}끼우다{\ quad} x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1130000sin(−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)+0.15⋅130000cos(−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)=73500
다듬다−102241.68342…=73500
⇒거짓
솔루션 확인 arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn:참
arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
n=1끼우다 arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1
130000sin(x)+0.15130000cos(x)=73500 위한 {\ quad}끼우다{\ quad} x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1130000sin(arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)+0.15⋅130000cos(arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)=73500
다듬다73500=73500
⇒참
솔루션 확인 2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn:거짓
2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
n=1끼우다 2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1
130000sin(x)+0.15130000cos(x)=73500 위한 {\ quad}끼우다{\ quad} x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1130000sin(2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)+0.15⋅130000cos(2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)=73500
다듬다−38288.87892…=73500
⇒거짓
x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
해를 10진수 형식으로 표시x=2.39936…+2πn,x=0.44444…+2πn