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인기 있는 삼각법 >

-5cos(x)+8.6602500000000…sin(x)=-5

해법

- 5 cos (x) + 8.6602500000000 \dots sin (x) = - 5

해법

x = - 2.09439 \dots + 2 πn, x = 0 + 2 πn, x = 2 π - 0 + 2 πn

+1

도

x = - 119.99997 \dots^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

솔루션 단계

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

더하다

5 cos (x)

양쪽으로

8.66025 \dots sin (x) = - 5 + 5 cos (x)

양쪽을 제곱

(8.66025 \dots sin (x))^{2} = (- 5 + 5 cos (x))^{2}

빼다

(- 5 + 5 cos (x))^{2}

양쪽에서

74.99993 \dots sin^{2} (x) - 25 + 50 cos (x) - 25 cos^{2} (x) = 0

삼각성을 사용하여 다시 쓰기

- 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots sin^{2} (x)

피타고라스 정체성 사용:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots (1 - cos^{2} (x))

- 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots (1 - cos^{2} (x))

간소화하다 :

50 cos (x) - 99.99993 \dots cos^{2} (x) + 49.99993 \dots

- 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots (1 - cos^{2} (x))

74.99993 \dots (1 - cos^{2} (x))

확대한다:

74.99993 \dots - 74.99993 \dots cos^{2} (x)

74.99993 \dots (1 - cos^{2} (x))

분배 법칙 적용:

a (b - c) = ab - a c

a = 74.99993 \dots, b = 1, c = cos^{2} (x)

= 74.99993 \dots \cdot 1 - 74.99993 \dots cos^{2} (x)

= 1 \cdot 74.99993 \dots - 74.99993 \dots cos^{2} (x)

숫자를 곱하시오:

1 \cdot 74.99993 \dots = 74.99993 \dots

= 74.99993 \dots - 74.99993 \dots cos^{2} (x)

= - 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots - 74.99993 \dots cos^{2} (x)

- 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots - 74.99993 \dots cos^{2} (x)

단순화하세요:

50 cos (x) - 99.99993 \dots cos^{2} (x) + 49.99993 \dots

- 25 - 25 cos^{2} (x) + 50 cos (x) + 74.99993 \dots - 74.99993 \dots cos^{2} (x)

집단적 용어

= - 25 cos^{2} (x) + 50 cos (x) - 74.99993 \dots cos^{2} (x) - 25 + 74.99993 \dots

유사 요소 추가:

- 25 cos^{2} (x) - 74.99993 \dots cos^{2} (x) = - 99.99993 \dots cos^{2} (x)

= - 99.99993 \dots cos^{2} (x) + 50 cos (x) - 25 + 74.99993 \dots

숫자 더하기/ 빼기:

- 25 + 74.99993 \dots = 49.99993 \dots

= 50 cos (x) - 99.99993 \dots cos^{2} (x) + 49.99993 \dots

= 50 cos (x) - 99.99993 \dots cos^{2} (x) + 49.99993 \dots

= 50 cos (x) - 99.99993 \dots cos^{2} (x) + 49.99993 \dots

49.99993 \dots + 50 cos (x) - 99.99993 \dots cos^{2} (x) = 0

대체로 해결

49.99993 \dots + 50 cos (x) - 99.99993 \dots cos^{2} (x) = 0

하게:

cos (x) = u

49.99993 \dots + 50 u - 99.99993 \dots u^{2} = 0

49.99993 \dots + 50 u - 99.99993 \dots u^{2} = 0 : u = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}, u = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

49.99993 \dots + 50 u - 99.99993 \dots u^{2} = 0

표준 양식으로 작성

a x^{2} + b x + c = 0

- 99.99993 \dots u^{2} + 50 u + 49.99993 \dots = 0

쿼드 공식으로 해결

- 99.99993 \dots u^{2} + 50 u + 49.99993 \dots = 0

4차 방정식 공식:

위해서

a = - 99.99993 \dots, b = 50, c = 49.99993 \dots

u_{1, 2} = \frac{- 50 \pm 5 0 ^{2} - 4 ( - 99.99993 \dots ) \cdot 49.99993 \dots}{2 ( - 99.99993 \dots )}

u_{1, 2} = \frac{- 50 \pm 5 0 ^{2} - 4 ( - 99.99993 \dots ) \cdot 49.99993 \dots}{2 ( - 99.99993 \dots )}

5 0^{2} - 4 (- 99.99993 \dots) \cdot 49.99993 \dots = 22499.95803 \dots

5 0^{2} - 4 (- 99.99993 \dots) \cdot 49.99993 \dots

규칙 적용

- (- a) = a

= 5 0^{2} + 4 \cdot 99.99993 \dots \cdot 49.99993 \dots

숫자를 곱하시오:

4 \cdot 99.99993 \dots \cdot 49.99993 \dots = 19999.95803 \dots

= 5 0^{2} + 19999.95803 \dots

5 0^{2} = 2500

= 2500 + 19999.95803 \dots

숫자 추가:

2500 + 19999.95803 \dots = 22499.95803 \dots

= 22499.95803 \dots

u_{1, 2} = \frac{- 50 \pm 22499.95803 \dots}{2 ( - 99.99993 \dots )}

솔루션 분리

u_{1} = \frac{- 50 + 22499.95803 \dots}{2 ( - 99.99993 \dots )}, u_{2} = \frac{- 50 - 22499.95803 \dots}{2 ( - 99.99993 \dots )}

u = \frac{- 50 + 22499.95803 \dots}{2 ( - 99.99993 \dots )} : - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}

\frac{- 50 + 22499.95803 \dots}{2 ( - 99.99993 \dots )}

괄호 제거:

(- a) = - a

= \frac{- 50 + 22499.95803 \dots}{- 2 \cdot 99.99993 \dots}

숫자를 곱하시오:

2 \cdot 99.99993 \dots = 199.99986 \dots

= \frac{- 50 + 22499.95803 \dots}{- 199.99986 \dots}

분수 규칙 적용:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}

u = \frac{- 50 - 22499.95803 \dots}{2 ( - 99.99993 \dots )} : \frac{50 + 22499.95803 \dots}{199.99986 \dots}

\frac{- 50 - 22499.95803 \dots}{2 ( - 99.99993 \dots )}

괄호 제거:

(- a) = - a

= \frac{- 50 - 22499.95803 \dots}{- 2 \cdot 99.99993 \dots}

숫자를 곱하시오:

2 \cdot 99.99993 \dots = 199.99986 \dots

= \frac{- 50 - 22499.95803 \dots}{- 199.99986 \dots}

분수 규칙 적용:

\frac{- a}{- b} = \frac{a}{b}

- 50 - 22499.95803 \dots = - (50 + 22499.95803 \dots)

= \frac{50 + 22499.95803 \dots}{199.99986 \dots}

2차 방정식의 해는 다음과 같다:

u = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}, u = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

뒤로 대체

u = cos (x)

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}, cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}, cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots} : x = arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}

트리거 역속성 적용

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}

일반 솔루션

cos (x) = - \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

x = arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots} : x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

트리거 역속성 적용

cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

일반 솔루션

cos (x) = \frac{50 + 22499.95803 \dots}{199.99986 \dots}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

모든 솔루션 결합

x = arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

해법을 원래 방정식에 연결하여 검증

솔루션을 에 연결하여 확인합니다

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

방정식에 맞지 않는 것은 제거하십시오.

솔루션 확인

arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn :

거짓

arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

n = 1

끼우다

arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

위한 {\ quad}끼우다{\ quad}

x = arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) + 8.66025 \dots sin (arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) = - 5

다듬다

9.99999 \dots = - 5

\Rightarrow 거짓

솔루션 확인

- arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn :

참

- arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

n = 1

끼우다

- arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

위한 {\ quad}끼우다{\ quad}

x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (- arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) + 8.66025 \dots sin (- arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) = - 5

다듬다

- 5 = - 5

\Rightarrow 참

솔루션 확인

arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn :

참

arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

n = 1

끼우다

arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

위한 {\ quad}끼우다{\ quad}

x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) + 8.66025 \dots sin (arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) = - 5

다듬다

- 5 = - 5

\Rightarrow 참

솔루션 확인

2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn :

참

2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

n = 1

끼우다

2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (x) + 8.66025 \dots sin (x) = - 5

위한 {\ quad}끼우다{\ quad}

x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1

- 5 cos (2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) + 8.66025 \dots sin (2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 π 1) = - 5

다듬다

- 5 = - 5

\Rightarrow 참

x = - arccos (- \frac{- 50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn, x = 2 π - arccos (\frac{50 + 22499.95803 \dots}{199.99986 \dots}) + 2 πn

해를 10진수 형식으로 표시

x = - 2.09439 \dots + 2 πn, x = 0 + 2 πn, x = 2 π - 0 + 2 πn

그래프

인기 있는 예

sin(2pi*1.5x)=0

sin (2 π \cdot 1.5 x) = 0

sin(15x)+cos(15x)=0

sin (15 x) + cos (15 x) = 0

sin(3θ)=-(sqrt(3))/2

sin (3 θ) = - \frac{3}{2}

solvefor x,arctan(y/x)=0

so l v e f or x, arctan (\frac{y}{x}) = 0

cos (x) = 3 sin (x)