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인기 있는 삼각법 >

9.8sin(x)-1.96cos(x)=8.54

해법

9.8 sin (x) - 1.96 cos (x) = 8.54

해법

x = 2.31438 \dots + 2 πn, x = 1.22199 \dots + 2 πn

+1

도

x = 132.60472 \dots^{\circ} + 36 0^{\circ} n, x = 70.01513 \dots^{\circ} + 36 0^{\circ} n

솔루션 단계

9.8 sin (x) - 1.96 cos (x) = 8.54

더하다

1.96 cos (x)

양쪽으로

9.8 sin (x) = 8.54 + 1.96 cos (x)

양쪽을 제곱

(9.8 sin (x))^{2} = (8.54 + 1.96 cos (x))^{2}

빼다

(8.54 + 1.96 cos (x))^{2}

양쪽에서

96.04 sin^{2} (x) - 72.9316 - 33.4768 cos (x) - 3.8416 cos^{2} (x) = 0

삼각성을 사용하여 다시 쓰기

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 sin^{2} (x)

피타고라스 정체성 사용:

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

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

= - 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x))

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x))

간소화하다 :

- 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x))

96.04 (1 - cos^{2} (x))

확대한다:

96.04 - 96.04 cos^{2} (x)

96.04 (1 - cos^{2} (x))

분배 법칙 적용:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (x)

= 96.04 \cdot 1 - 96.04 cos^{2} (x)

= 1 \cdot 96.04 - 96.04 cos^{2} (x)

숫자를 곱하시오:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (x)

= - 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x)

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x)

단순화하세요:

- 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x)

집단적 용어

= - 3.8416 cos^{2} (x) - 33.4768 cos (x) - 96.04 cos^{2} (x) - 72.9316 + 96.04

유사 요소 추가:

- 3.8416 cos^{2} (x) - 96.04 cos^{2} (x) = - 99.8816 cos^{2} (x)

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) - 72.9316 + 96.04

숫자 더하기/ 빼기:

- 72.9316 + 96.04 = 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

23.1084 - 33.4768 cos (x) - 99.8816 cos^{2} (x) = 0

대체로 해결

23.1084 - 33.4768 cos (x) - 99.8816 cos^{2} (x) = 0

하게:

cos (x) = u

23.1084 - 33.4768 u - 99.8816 u^{2} = 0

23.1084 - 33.4768 u - 99.8816 u^{2} = 0 : u = - \frac{33.4768 + 10353.112}{199.7632}, u = \frac{10353.112 - 33.4768}{199.7632}

23.1084 - 33.4768 u - 99.8816 u^{2} = 0

표준 양식으로 작성

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

- 99.8816 u^{2} - 33.4768 u + 23.1084 = 0

쿼드 공식으로 해결

- 99.8816 u^{2} - 33.4768 u + 23.1084 = 0

4차 방정식 공식:

위해서

a = - 99.8816, b = - 33.4768, c = 23.1084

u_{1, 2} = \frac{- ( - 33.4768 ) \pm ( - 33.4768 ) ^{2} - 4 ( - 99.8816 ) \cdot 23.1084}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 33.4768 ) \pm ( - 33.4768 ) ^{2} - 4 ( - 99.8816 ) \cdot 23.1084}{2 ( - 99.8816 )}

(- 33.4768)^{2} - 4 (- 99.8816) \cdot 23.1084 = 10353.112

(- 33.4768)^{2} - 4 (- 99.8816) \cdot 23.1084

규칙 적용

- (- a) = a

= (- 33.4768)^{2} + 4 \cdot 99.8816 \cdot 23.1084

지수 규칙 적용:

(- a)^{n} = a^{n},

이면

n

균등하다

(- 33.4768)^{2} = 33.476 8^{2}

= 33.476 8^{2} + 4 \cdot 23.1084 \cdot 99.8816

숫자를 곱하시오:

4 \cdot 99.8816 \cdot 23.1084 = 9232.41586 \dots

= 33.476 8^{2} + 9232.41586 \dots

33.476 8^{2} = 1120.69613 \dots

= 1120.69613 \dots + 9232.41586 \dots

숫자 추가:

1120.69613 \dots + 9232.41586 \dots = 10353.112

= 10353.112

u_{1, 2} = \frac{- ( - 33.4768 ) \pm 10353.112}{2 ( - 99.8816 )}

솔루션 분리

u_{1} = \frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )}

u = \frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )} : - \frac{33.4768 + 10353.112}{199.7632}

\frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )}

괄호 제거:

(- a) = - a, - (- a) = a

= \frac{33.4768 + 10353.112}{- 2 \cdot 99.8816}

숫자를 곱하시오:

2 \cdot 99.8816 = 199.7632

= \frac{33.4768 + 10353.112}{- 199.7632}

분수 규칙 적용:

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

= - \frac{33.4768 + 10353.112}{199.7632}

u = \frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )} : \frac{10353.112 - 33.4768}{199.7632}

\frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )}

괄호 제거:

(- a) = - a, - (- a) = a

= \frac{33.4768 - 10353.112}{- 2 \cdot 99.8816}

숫자를 곱하시오:

2 \cdot 99.8816 = 199.7632

= \frac{33.4768 - 10353.112}{- 199.7632}

분수 규칙 적용:

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

33.4768 - 10353.112 = - (10353.112 - 33.4768)

= \frac{10353.112 - 33.4768}{199.7632}

2차 방정식의 해는 다음과 같다:

u = - \frac{33.4768 + 10353.112}{199.7632}, u = \frac{10353.112 - 33.4768}{199.7632}

뒤로 대체

u = cos (x)

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}, cos (x) = \frac{10353.112 - 33.4768}{199.7632}

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}, cos (x) = \frac{10353.112 - 33.4768}{199.7632}

cos (x) = - \frac{33.4768 + 10353.112}{199.7632} : x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

트리거 역속성 적용

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

일반 솔루션

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

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

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

cos (x) = \frac{10353.112 - 33.4768}{199.7632} : x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

트리거 역속성 적용

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

일반 솔루션

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

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

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

모든 솔루션 결합

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

해법을 원래 방정식에 연결하여 검증

솔루션을 에 연결하여 확인합니다

9.8 sin (x) - 1.96 cos (x) = 8.54

방정식에 맞지 않는 것은 제거하십시오.

솔루션 확인

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn :

참

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

n = 1

끼우다

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (x) - 1.96 cos (x) = 8.54

위한 {\ quad}끼우다{\ quad}

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) = 8.54

다듬다

8.54 = 8.54

\Rightarrow 참

솔루션 확인

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn :

거짓

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

n = 1

끼우다

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (x) - 1.96 cos (x) = 8.54

위한 {\ quad}끼우다{\ quad}

x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) = 8.54

다듬다

- 5.88640 \dots = 8.54

\Rightarrow 거짓

솔루션 확인

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn :

참

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

n = 1

끼우다

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (x) - 1.96 cos (x) = 8.54

위한 {\ quad}끼우다{\ quad}

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) = 8.54

다듬다

8.54 = 8.54

\Rightarrow 참

솔루션 확인

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn :

거짓

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

n = 1

끼우다

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (x) - 1.96 cos (x) = 8.54

위한 {\ quad}끼우다{\ quad}

x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) = 8.54

다듬다

- 9.87974 \dots = 8.54

\Rightarrow 거짓

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

해를 10진수 형식으로 표시

x = 2.31438 \dots + 2 πn, x = 1.22199 \dots + 2 πn

그래프

인기 있는 예

3/(cot(x))=8-5tan(x)

\frac{3}{cot ( x )} = 8 - 5 tan (x)

sin(θ)= 3/8 ,sin(2θ)

sin (θ) = \frac{3}{8}, sin (2 θ)

2(1-cos^2(x))= 3/2

2 (1 - cos^{2} (x)) = \frac{3}{2}

cos(3θ)=-(sqrt(2))/2

cos (3 θ) = - \frac{2}{2}

25*sin(x)-1.5*cos(x)=20

25 \cdot sin (x) - 1.5 \cdot cos (x) = 20