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인기 있는 삼각법 >

9.8sin(α)-1.96cos(α)=0.47

해법

9.8 \cdot sin (α) - 1.96 \cdot cos (α) = 0.47

해법

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

+1

도

α = - 171.38555 \dots^{\circ} + 36 0^{\circ} n, α = 14.00542 \dots^{\circ} + 36 0^{\circ} n

솔루션 단계

9.8 sin (α) - 1.96 cos (α) = 0.47

더하다

1.96 cos (α)

양쪽으로

9.8 sin (α) = 0.47 + 1.96 cos (α)

양쪽을 제곱

(9.8 sin (α))^{2} = (0.47 + 1.96 cos (α))^{2}

빼다

(0.47 + 1.96 cos (α))^{2}

양쪽에서

96.04 sin^{2} (α) - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) = 0

삼각성을 사용하여 다시 쓰기

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 sin^{2} (α)

피타고라스 정체성 사용:

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

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

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

간소화하다 :

- 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

96.04 (1 - cos^{2} (α))

확대한다:

96.04 - 96.04 cos^{2} (α)

96.04 (1 - cos^{2} (α))

분배 법칙 적용:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (α)

= 96.04 \cdot 1 - 96.04 cos^{2} (α)

= 1 \cdot 96.04 - 96.04 cos^{2} (α)

숫자를 곱하시오:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (α)

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

단순화하세요:

- 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

집단적 용어

= - 1.8424 cos (α) - 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) - 0.2209 + 96.04

유사 요소 추가:

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

= - 1.8424 cos (α) - 99.8816 cos^{2} (α) - 0.2209 + 96.04

숫자 더하기/ 빼기:

- 0.2209 + 96.04 = 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

대체로 해결

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

하게:

cos (α) = u

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

95.8191 - 1.8424 u - 99.8816 u^{2} = 0 : u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

표준 양식으로 작성

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

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

쿼드 공식으로 해결

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

4차 방정식 공식:

위해서

a = - 99.8816, b = - 1.8424, c = 95.8191

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191 = 38285.654512

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191

규칙 적용

- (- a) = a

= (- 1.8424)^{2} + 4 \cdot 99.8816 \cdot 95.8191

지수 규칙 적용:

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

이면

n

균등하다

(- 1.8424)^{2} = 1.842 4^{2}

= 1.842 4^{2} + 4 \cdot 95.8191 \cdot 99.8816

숫자를 곱하시오:

4 \cdot 99.8816 \cdot 95.8191 = 38282.26007 \dots

= 1.842 4^{2} + 38282.26007 \dots

1.842 4^{2} = 3.39443776

= 3.39443776 + 38282.26007 \dots

숫자 추가:

3.39443776 + 38282.26007 \dots = 38285.654512

= 38285.654512

u_{1, 2} = \frac{- ( - 1.8424 ) \pm 38285.654512}{2 ( - 99.8816 )}

솔루션 분리

u_{1} = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

u = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )} : - \frac{1.8424 + 38285.654512}{199.7632}

\frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}

괄호 제거:

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

= \frac{1.8424 + 38285.654512}{- 2 \cdot 99.8816}

숫자를 곱하시오:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 + 38285.654512}{- 199.7632}

분수 규칙 적용:

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

= - \frac{1.8424 + 38285.654512}{199.7632}

u = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )} : \frac{38285.654512 - 1.8424}{199.7632}

\frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

괄호 제거:

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

= \frac{1.8424 - 38285.654512}{- 2 \cdot 99.8816}

숫자를 곱하시오:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 - 38285.654512}{- 199.7632}

분수 규칙 적용:

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

1.8424 - 38285.654512 = - (38285.654512 - 1.8424)

= \frac{38285.654512 - 1.8424}{199.7632}

2차 방정식의 해는 다음과 같다:

u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

뒤로 대체

u = cos (α)

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632} : α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

트리거 역속성 적용

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

일반 솔루션

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

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

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632} : α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

트리거 역속성 적용

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

일반 솔루션

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

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

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

모든 솔루션 결합

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

해법을 원래 방정식에 연결하여 검증

솔루션을 에 연결하여 확인합니다

9.8 sin (α) - 1.96 cos (α) = 0.47

방정식에 맞지 않는 것은 제거하십시오.

솔루션 확인

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn :

거짓

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

n = 1

끼우다

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (α) - 1.96 cos (α) = 0.47

위한 {\ quad}끼우다{\ quad}

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

다듬다

3.40577 \dots = 0.47

\Rightarrow 거짓

솔루션 확인

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn :

참

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

n = 1

끼우다

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (α) - 1.96 cos (α) = 0.47

위한 {\ quad}끼우다{\ quad}

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

다듬다

0.47 = 0.47

\Rightarrow 참

솔루션 확인

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn :

참

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

n = 1

끼우다

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (α) - 1.96 cos (α) = 0.47

위한 {\ quad}끼우다{\ quad}

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

다듬다

0.47 = 0.47

\Rightarrow 참

솔루션 확인

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn :

거짓

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

n = 1

끼우다

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (α) - 1.96 cos (α) = 0.47

위한 {\ quad}끼우다{\ quad}

α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

다듬다

- 4.27346 \dots = 0.47

\Rightarrow 거짓

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

해를 10진수 형식으로 표시

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

그래프

인기 있는 예

cos(pi+x)= 1/(sqrt(2))

cos (π + x) = \frac{1}{2}

2 sin (x) = sec (x)

7(sin(x))(cos(x))+cos^2(x)=1

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

2sec(x)=tan(x)+cot(x)

2 sec (x) = tan (x) + cot (x)

sqrt(3)sin(2x)=cos(2x)

3 sin (2 x) = cos (2 x)