해법
229cos(a)+14715sin(a)=3959.55
해법
a=0.25684…+2πn,a=π−0.28796…+2πn
+1
도
a=14.71615…∘+360∘n,a=163.50067…∘+360∘n솔루션 단계
229cos(a)+14715sin(a)=3959.55
빼다 14715sin(a) 양쪽에서229cos(a)=3959.55−14715sin(a)
양쪽을 제곱(229cos(a))2=(3959.55−14715sin(a))2
빼다 (3959.55−14715sin(a))2 양쪽에서52441cos2(a)−15678036.2025+116529556.5sin(a)−216531225sin2(a)=0
삼각성을 사용하여 다시 쓰기
−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441cos2(a)
피타고라스 정체성 사용: cos2(x)+sin2(x)=1cos2(x)=1−sin2(x)=−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441(1−sin2(a))
−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441(1−sin2(a))간소화하다 :116529556.5sin(a)−216583666sin2(a)−15625595.2025
−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441(1−sin2(a))
52441(1−sin2(a))확대한다:52441−52441sin2(a)
52441(1−sin2(a))
분배 법칙 적용: a(b−c)=ab−aca=52441,b=1,c=sin2(a)=52441⋅1−52441sin2(a)
숫자를 곱하시오: 52441⋅1=52441=52441−52441sin2(a)
=−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441−52441sin2(a)
−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441−52441sin2(a)단순화하세요:116529556.5sin(a)−216583666sin2(a)−15625595.2025
−15678036.2025+116529556.5sin(a)−216531225sin2(a)+52441−52441sin2(a)
집단적 용어=116529556.5sin(a)−216531225sin2(a)−52441sin2(a)−15678036.2025+52441
유사 요소 추가: −216531225sin2(a)−52441sin2(a)=−216583666sin2(a)=116529556.5sin(a)−216583666sin2(a)−15678036.2025+52441
숫자 더하기/ 빼기: −15678036.2025+52441=−15625595.2025=116529556.5sin(a)−216583666sin2(a)−15625595.2025
=116529556.5sin(a)−216583666sin2(a)−15625595.2025
=116529556.5sin(a)−216583666sin2(a)−15625595.2025
−15625595.2025+116529556.5sin(a)−216583666sin2(a)=0
대체로 해결
−15625595.2025+116529556.5sin(a)−216583666sin2(a)=0
하게: sin(a)=u−15625595.2025+116529556.5u−216583666u2=0
−15625595.2025+116529556.5u−216583666u2=0:u=433167332−116529556.52−1.3537E16+116529556.5,u=433167332116529556.52−1.3537E16+116529556.5
−15625595.2025+116529556.5u−216583666u2=0
표준 양식으로 작성 ax2+bx+c=0−216583666u2+116529556.5u−15625595.2025=0
쿼드 공식으로 해결
−216583666u2+116529556.5u−15625595.2025=0
4차 방정식 공식:
위해서 a=−216583666,b=116529556.5,c=−15625595.2025u1,2=2(−216583666)−116529556.5±116529556.52−4(−216583666)(−15625595.2025)
u1,2=2(−216583666)−116529556.5±116529556.52−4(−216583666)(−15625595.2025)
116529556.52−4(−216583666)(−15625595.2025)=116529556.52−1.3537E16
116529556.52−4(−216583666)(−15625595.2025)
규칙 적용 −(−a)=a=116529556.52−4⋅216583666⋅15625595.2025
숫자를 곱하시오: 4⋅216583666⋅15625595.2025=1.3537E16=116529556.52−1.3537E16
u1,2=2(−216583666)−116529556.5±116529556.52−1.3537E16
솔루션 분리u1=2(−216583666)−116529556.5+116529556.52−1.3537E16,u2=2(−216583666)−116529556.5−116529556.52−1.3537E16
u=2(−216583666)−116529556.5+116529556.52−1.3537E16:433167332−116529556.52−1.3537E16+116529556.5
2(−216583666)−116529556.5+116529556.52−1.3537E16
괄호 제거: (−a)=−a=−2⋅216583666−116529556.5+116529556.52−1.3537E16
숫자를 곱하시오: 2⋅216583666=433167332=−433167332−116529556.5+116529556.52−1.3537E16
분수 규칙 적용: −b−a=ba−116529556.5+116529556.52−1.3537E16=−(−116529556.52−1.3537E16+116529556.5)=433167332−116529556.52−1.3537E16+116529556.5
u=2(−216583666)−116529556.5−116529556.52−1.3537E16:433167332116529556.52−1.3537E16+116529556.5
2(−216583666)−116529556.5−116529556.52−1.3537E16
괄호 제거: (−a)=−a=−2⋅216583666−116529556.5−116529556.52−1.3537E16
숫자를 곱하시오: 2⋅216583666=433167332=−433167332−116529556.5−116529556.52−1.3537E16
분수 규칙 적용: −b−a=ba−116529556.5−116529556.52−1.3537E16=−(116529556.52−1.3537E16+116529556.5)=433167332116529556.52−1.3537E16+116529556.5
2차 방정식의 해는 다음과 같다:u=433167332−116529556.52−1.3537E16+116529556.5,u=433167332116529556.52−1.3537E16+116529556.5
뒤로 대체 u=sin(a)sin(a)=433167332−116529556.52−1.3537E16+116529556.5,sin(a)=433167332116529556.52−1.3537E16+116529556.5
sin(a)=433167332−116529556.52−1.3537E16+116529556.5,sin(a)=433167332116529556.52−1.3537E16+116529556.5
sin(a)=433167332−116529556.52−1.3537E16+116529556.5:a=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn
sin(a)=433167332−116529556.52−1.3537E16+116529556.5
트리거 역속성 적용
sin(a)=433167332−116529556.52−1.3537E16+116529556.5
일반 솔루션 sin(a)=433167332−116529556.52−1.3537E16+116529556.5sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πna=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn
a=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn
sin(a)=433167332116529556.52−1.3537E16+116529556.5:a=arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
sin(a)=433167332116529556.52−1.3537E16+116529556.5
트리거 역속성 적용
sin(a)=433167332116529556.52−1.3537E16+116529556.5
일반 솔루션 sin(a)=433167332116529556.52−1.3537E16+116529556.5sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πna=arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
a=arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
모든 솔루션 결합a=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
해법을 원래 방정식에 연결하여 검증
솔루션을 에 연결하여 확인합니다 229cos(a)+14715sin(a)=3959.55
방정식에 맞지 않는 것은 제거하십시오.
솔루션 확인 arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn:참
arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn
n=1끼우다 arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1
229cos(a)+14715sin(a)=3959.55 위한 {\ quad}끼우다{\ quad} a=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1229cos(arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1)+14715sin(arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1)=3959.55
다듬다3959.54999…=3959.55
⇒참
솔루션 확인 π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn:거짓
π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn
n=1끼우다 π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1
229cos(a)+14715sin(a)=3959.55 위한 {\ quad}끼우다{\ quad} a=π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1229cos(π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1)+14715sin(π−arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2π1)=3959.55
다듬다3516.57416…=3959.55
⇒거짓
솔루션 확인 arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn:거짓
arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
n=1끼우다 arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1
229cos(a)+14715sin(a)=3959.55 위한 {\ quad}끼우다{\ quad} a=arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1229cos(arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1)+14715sin(arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1)=3959.55
다듬다4398.69096…=3959.55
⇒거짓
솔루션 확인 π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn:참
π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
n=1끼우다 π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1
229cos(a)+14715sin(a)=3959.55 위한 {\ quad}끼우다{\ quad} a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1229cos(π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1)+14715sin(π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2π1)=3959.55
다듬다3959.54999…=3959.55
⇒참
a=arcsin(433167332−116529556.52−1.3537E16+116529556.5)+2πn,a=π−arcsin(433167332116529556.52−1.3537E16+116529556.5)+2πn
해를 10진수 형식으로 표시a=0.25684…+2πn,a=π−0.28796…+2πn