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인기 있는 삼각법 >

sin(θ)=0.6367+0.1*cos(θ)

해법

sin (θ) = 0.6367 + 0.1 \cdot cos (θ)

해법

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

+1

도

θ = 146.39879 \dots^{\circ} + 36 0^{\circ} n, θ = 45.02238 \dots^{\circ} + 36 0^{\circ} n

솔루션 단계

sin (θ) = 0.6367 + 0.1 cos (θ)

양쪽을 제곱

sin^{2} (θ) = (0.6367 + 0.1 cos (θ))^{2}

빼다

(0.6367 + 0.1 cos (θ))^{2}

양쪽에서

sin^{2} (θ) - 0.40538689 - 0.12734 cos (θ) - 0.01 cos^{2} (θ) = 0

삼각성을 사용하여 다시 쓰기

- 0.40538689 + sin^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

피타고라스 정체성 사용:

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

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

= - 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

- 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

간소화하다 :

- 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

- 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

유사 요소 추가:

- cos^{2} (θ) - 0.01 cos^{2} (θ) = - 1.01 cos^{2} (θ)

= - 0.40538689 + 1 - 1.01 cos^{2} (θ) - 0.12734 cos (θ)

숫자 더하기/ 빼기:

- 0.40538689 + 1 = 0.59461311

= - 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

= - 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

0.59461311 - 0.12734 cos (θ) - 1.01 cos^{2} (θ) = 0

대체로 해결

0.59461311 - 0.12734 cos (θ) - 1.01 cos^{2} (θ) = 0

하게:

cos (θ) = u

0.59461311 - 0.12734 u - 1.01 u^{2} = 0

0.59461311 - 0.12734 u - 1.01 u^{2} = 0 : u = - \frac{0.12734 + 2.41845244}{2.02}, u = \frac{2.41845244 - 0.12734}{2.02}

0.59461311 - 0.12734 u - 1.01 u^{2} = 0

표준 양식으로 작성

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

- 1.01 u^{2} - 0.12734 u + 0.59461311 = 0

쿼드 공식으로 해결

- 1.01 u^{2} - 0.12734 u + 0.59461311 = 0

4차 방정식 공식:

위해서

a = - 1.01, b = - 0.12734, c = 0.59461311

u_{1, 2} = \frac{- ( - 0.12734 ) \pm ( - 0.12734 ) ^{2} - 4 ( - 1.01 ) \cdot 0.59461311}{2 ( - 1.01 )}

u_{1, 2} = \frac{- ( - 0.12734 ) \pm ( - 0.12734 ) ^{2} - 4 ( - 1.01 ) \cdot 0.59461311}{2 ( - 1.01 )}

(- 0.12734)^{2} - 4 (- 1.01) \cdot 0.59461311 = 2.41845244

(- 0.12734)^{2} - 4 (- 1.01) \cdot 0.59461311

규칙 적용

- (- a) = a

= (- 0.12734)^{2} + 4 \cdot 1.01 \cdot 0.59461311

지수 규칙 적용:

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

이면

n

균등하다

(- 0.12734)^{2} = 0.1273 4^{2}

= 0.1273 4^{2} + 4 \cdot 0.59461311 \cdot 1.01

숫자를 곱하시오:

4 \cdot 1.01 \cdot 0.59461311 = 2.40223 \dots

= 0.1273 4^{2} + 2.40223 \dots

0.1273 4^{2} = 0.0162154756

= 0.0162154756 + 2.40223 \dots

숫자 추가:

0.0162154756 + 2.40223 \dots = 2.41845244

= 2.41845244

u_{1, 2} = \frac{- ( - 0.12734 ) \pm 2.41845244}{2 ( - 1.01 )}

솔루션 분리

u_{1} = \frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )}, u_{2} = \frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )}

u = \frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )} : - \frac{0.12734 + 2.41845244}{2.02}

\frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )}

괄호 제거:

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

= \frac{0.12734 + 2.41845244}{- 2 \cdot 1.01}

숫자를 곱하시오:

2 \cdot 1.01 = 2.02

= \frac{0.12734 + 2.41845244}{- 2.02}

분수 규칙 적용:

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

= - \frac{0.12734 + 2.41845244}{2.02}

u = \frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )} : \frac{2.41845244 - 0.12734}{2.02}

\frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )}

괄호 제거:

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

= \frac{0.12734 - 2.41845244}{- 2 \cdot 1.01}

숫자를 곱하시오:

2 \cdot 1.01 = 2.02

= \frac{0.12734 - 2.41845244}{- 2.02}

분수 규칙 적용:

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

0.12734 - 2.41845244 = - (2.41845244 - 0.12734)

= \frac{2.41845244 - 0.12734}{2.02}

2차 방정식의 해는 다음과 같다:

u = - \frac{0.12734 + 2.41845244}{2.02}, u = \frac{2.41845244 - 0.12734}{2.02}

뒤로 대체

u = cos (θ)

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}, cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}, cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02} : θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

트리거 역속성 적용

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

일반 솔루션

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

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

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

cos (θ) = \frac{2.41845244 - 0.12734}{2.02} : θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

트리거 역속성 적용

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

일반 솔루션

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

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

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

모든 솔루션 결합

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

해법을 원래 방정식에 연결하여 검증

솔루션을 에 연결하여 확인합니다

sin (θ) = 0.6367 + 0.1 cos (θ)

방정식에 맞지 않는 것은 제거하십시오.

솔루션 확인

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn :

참

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

n = 1

끼우다

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (θ) = 0.6367 + 0.1 cos (θ)

위한 {\ quad}끼우다{\ quad}

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1)

다듬다

0.55340 \dots = 0.55340 \dots

\Rightarrow 참

솔루션 확인

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn :

거짓

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

n = 1

끼우다

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (θ) = 0.6367 + 0.1 cos (θ)

위한 {\ quad}끼우다{\ quad}

θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1)

다듬다

- 0.55340 \dots = 0.55340 \dots

\Rightarrow 거짓

솔루션 확인

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn :

참

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

n = 1

끼우다

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (θ) = 0.6367 + 0.1 cos (θ)

위한 {\ quad}끼우다{\ quad}

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1)

다듬다

0.70738 \dots = 0.70738 \dots

\Rightarrow 참

솔루션 확인

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn :

거짓

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

n = 1

끼우다

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (θ) = 0.6367 + 0.1 cos (θ)

위한 {\ quad}끼우다{\ quad}

θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1)

다듬다

- 0.70738 \dots = 0.70738 \dots

\Rightarrow 거짓

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

해를 10진수 형식으로 표시

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

그래프

인기 있는 예

sec(x)+tan(x)=sqrt(3)

sec (x) + tan (x) = 3

cos (x) = \frac{10}{21}

sqrt(3)tan^2(θ)+2tan(θ)-sqrt(3)=0

3 tan^{2} (θ) + 2 tan (θ) - 3 = 0

sec (x) = \frac{25}{7}

sin(x/2)= 1/2 sin(x)

sin (\frac{x}{2}) = \frac{1}{2} sin (x)