해법
−11cos(x)22−11sin(x)8=−250
해법
x=0.10677…+2πn,x=0.59076…+2πn
+1
도
x=6.11761…∘+360∘n,x=33.84860…∘+360∘n솔루션 단계
−11cos(x)⋅22−11sin(x)⋅8=−250
더하다 11sin(x)8 양쪽으로−242cos(x)=−250+88sin(x)
양쪽을 제곱(−242cos(x))2=(−250+88sin(x))2
빼다 (−250+88sin(x))2 양쪽에서58564cos2(x)−62500+44000sin(x)−7744sin2(x)=0
삼각성을 사용하여 다시 쓰기
−62500+44000sin(x)+58564cos2(x)−7744sin2(x)
피타고라스 정체성 사용: cos2(x)+sin2(x)=1cos2(x)=1−sin2(x)=−62500+44000sin(x)+58564(1−sin2(x))−7744sin2(x)
−62500+44000sin(x)+58564(1−sin2(x))−7744sin2(x)간소화하다 :44000sin(x)−66308sin2(x)−3936
−62500+44000sin(x)+58564(1−sin2(x))−7744sin2(x)
58564(1−sin2(x))확대한다:58564−58564sin2(x)
58564(1−sin2(x))
분배 법칙 적용: a(b−c)=ab−aca=58564,b=1,c=sin2(x)=58564⋅1−58564sin2(x)
숫자를 곱하시오: 58564⋅1=58564=58564−58564sin2(x)
=−62500+44000sin(x)+58564−58564sin2(x)−7744sin2(x)
−62500+44000sin(x)+58564−58564sin2(x)−7744sin2(x)단순화하세요:44000sin(x)−66308sin2(x)−3936
−62500+44000sin(x)+58564−58564sin2(x)−7744sin2(x)
유사 요소 추가: −58564sin2(x)−7744sin2(x)=−66308sin2(x)=−62500+44000sin(x)+58564−66308sin2(x)
집단적 용어=44000sin(x)−66308sin2(x)−62500+58564
숫자 더하기/ 빼기: −62500+58564=−3936=44000sin(x)−66308sin2(x)−3936
=44000sin(x)−66308sin2(x)−3936
=44000sin(x)−66308sin2(x)−3936
−3936+44000sin(x)−66308sin2(x)=0
대체로 해결
−3936+44000sin(x)−66308sin2(x)=0
하게: sin(x)=u−3936+44000u−66308u2=0
−3936+44000u−66308u2=0:u=132616−440002−1043953152+44000,u=132616440002−1043953152+44000
−3936+44000u−66308u2=0
표준 양식으로 작성 ax2+bx+c=0−66308u2+44000u−3936=0
쿼드 공식으로 해결
−66308u2+44000u−3936=0
4차 방정식 공식:
위해서 a=−66308,b=44000,c=−3936u1,2=2(−66308)−44000±440002−4(−66308)(−3936)
u1,2=2(−66308)−44000±440002−4(−66308)(−3936)
440002−4(−66308)(−3936)=440002−1043953152
440002−4(−66308)(−3936)
규칙 적용 −(−a)=a=440002−4⋅66308⋅3936
숫자를 곱하시오: 4⋅66308⋅3936=1043953152=440002−1043953152
u1,2=2(−66308)−44000±440002−1043953152
솔루션 분리u1=2(−66308)−44000+440002−1043953152,u2=2(−66308)−44000−440002−1043953152
u=2(−66308)−44000+440002−1043953152:132616−440002−1043953152+44000
2(−66308)−44000+440002−1043953152
괄호 제거: (−a)=−a=−2⋅66308−44000+440002−1043953152
숫자를 곱하시오: 2⋅66308=132616=−132616−44000+440002−1043953152
분수 규칙 적용: −b−a=ba−44000+440002−1043953152=−(−440002−1043953152+44000)=132616−440002−1043953152+44000
u=2(−66308)−44000−440002−1043953152:132616440002−1043953152+44000
2(−66308)−44000−440002−1043953152
괄호 제거: (−a)=−a=−2⋅66308−44000−440002−1043953152
숫자를 곱하시오: 2⋅66308=132616=−132616−44000−440002−1043953152
분수 규칙 적용: −b−a=ba−44000−440002−1043953152=−(440002−1043953152+44000)=132616440002−1043953152+44000
2차 방정식의 해는 다음과 같다:u=132616−440002−1043953152+44000,u=132616440002−1043953152+44000
뒤로 대체 u=sin(x)sin(x)=132616−440002−1043953152+44000,sin(x)=132616440002−1043953152+44000
sin(x)=132616−440002−1043953152+44000,sin(x)=132616440002−1043953152+44000
sin(x)=132616−440002−1043953152+44000:x=arcsin(132616−440002−1043953152+44000)+2πn,x=π−arcsin(132616−440002−1043953152+44000)+2πn
sin(x)=132616−440002−1043953152+44000
트리거 역속성 적용
sin(x)=132616−440002−1043953152+44000
일반 솔루션 sin(x)=132616−440002−1043953152+44000sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πnx=arcsin(132616−440002−1043953152+44000)+2πn,x=π−arcsin(132616−440002−1043953152+44000)+2πn
x=arcsin(132616−440002−1043953152+44000)+2πn,x=π−arcsin(132616−440002−1043953152+44000)+2πn
sin(x)=132616440002−1043953152+44000:x=arcsin(132616440002−1043953152+44000)+2πn,x=π−arcsin(132616440002−1043953152+44000)+2πn
sin(x)=132616440002−1043953152+44000
트리거 역속성 적용
sin(x)=132616440002−1043953152+44000
일반 솔루션 sin(x)=132616440002−1043953152+44000sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πnx=arcsin(132616440002−1043953152+44000)+2πn,x=π−arcsin(132616440002−1043953152+44000)+2πn
x=arcsin(132616440002−1043953152+44000)+2πn,x=π−arcsin(132616440002−1043953152+44000)+2πn
모든 솔루션 결합x=arcsin(132616−440002−1043953152+44000)+2πn,x=π−arcsin(132616−440002−1043953152+44000)+2πn,x=arcsin(132616440002−1043953152+44000)+2πn,x=π−arcsin(132616440002−1043953152+44000)+2πn
해법을 원래 방정식에 연결하여 검증
솔루션을 에 연결하여 확인합니다 −11cos(x)22−11sin(x)8=−250
방정식에 맞지 않는 것은 제거하십시오.
솔루션 확인 arcsin(132616−440002−1043953152+44000)+2πn:참
arcsin(132616−440002−1043953152+44000)+2πn
n=1끼우다 arcsin(132616−440002−1043953152+44000)+2π1
−11cos(x)22−11sin(x)8=−250 위한 {\ quad}끼우다{\ quad} x=arcsin(132616−440002−1043953152+44000)+2π1−11cos(arcsin(132616−440002−1043953152+44000)+2π1)⋅22−11sin(arcsin(132616−440002−1043953152+44000)+2π1)⋅8=−250
다듬다−250=−250
⇒참
솔루션 확인 π−arcsin(132616−440002−1043953152+44000)+2πn:거짓
π−arcsin(132616−440002−1043953152+44000)+2πn
n=1끼우다 π−arcsin(132616−440002−1043953152+44000)+2π1
−11cos(x)22−11sin(x)8=−250 위한 {\ quad}끼우다{\ quad} x=π−arcsin(132616−440002−1043953152+44000)+2π1−11cos(π−arcsin(132616−440002−1043953152+44000)+2π1)⋅22−11sin(π−arcsin(132616−440002−1043953152+44000)+2π1)⋅8=−250
다듬다231.24373…=−250
⇒거짓
솔루션 확인 arcsin(132616440002−1043953152+44000)+2πn:참
arcsin(132616440002−1043953152+44000)+2πn
n=1끼우다 arcsin(132616440002−1043953152+44000)+2π1
−11cos(x)22−11sin(x)8=−250 위한 {\ quad}끼우다{\ quad} x=arcsin(132616440002−1043953152+44000)+2π1−11cos(arcsin(132616440002−1043953152+44000)+2π1)⋅22−11sin(arcsin(132616440002−1043953152+44000)+2π1)⋅8=−250
다듬다−250=−250
⇒참
솔루션 확인 π−arcsin(132616440002−1043953152+44000)+2πn:거짓
π−arcsin(132616440002−1043953152+44000)+2πn
n=1끼우다 π−arcsin(132616440002−1043953152+44000)+2π1
−11cos(x)22−11sin(x)8=−250 위한 {\ quad}끼우다{\ quad} x=π−arcsin(132616440002−1043953152+44000)+2π1−11cos(π−arcsin(132616440002−1043953152+44000)+2π1)⋅22−11sin(π−arcsin(132616440002−1043953152+44000)+2π1)⋅8=−250
다듬다151.96794…=−250
⇒거짓
x=arcsin(132616−440002−1043953152+44000)+2πn,x=arcsin(132616440002−1043953152+44000)+2πn
해를 10진수 형식으로 표시x=0.10677…+2πn,x=0.59076…+2πn