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인기 있는 삼각법 >

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

해법

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

해법

x = - 0.86437 \dots + 2 πn, x = π + 0.86437 \dots + 2 πn

+1

도

x = - 49.52505 \dots^{\circ} + 36 0^{\circ} n, x = 229.52505 \dots^{\circ} + 36 0^{\circ} n

솔루션 단계

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

빼다

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

양쪽에서

sin (x) cos (2 x) - \frac{1}{2} (1 + sin (x)) = 0

sin (x) cos (2 x) - \frac{1}{2} (1 + sin (x))

단순화하세요:

\frac{2 sin ( x ) cos ( 2 x ) - 1 - sin ( x )}{2}

sin (x) cos (2 x) - \frac{1}{2} (1 + sin (x))

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

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

다중 분수:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

1 \cdot (1 + sin (x)) = 1 + sin (x)

1 \cdot (1 + sin (x))

곱하다:

1 \cdot (1 + sin (x)) = (1 + sin (x))

= (1 + sin (x))

괄호 제거:

(a) = a

= 1 + sin (x)

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

= sin (x) cos (2 x) - \frac{sin ( x ) + 1}{2}

요소를 분수로 변환:

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

= \frac{sin ( x ) cos ( 2 x ) \cdot 2}{2} - \frac{1 + sin ( x )}{2}

분모가 같기 때문에, 분수를 합친다:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( x ) cos ( 2 x ) \cdot 2 - ( 1 + sin ( x ) )}{2}

sin (x) cos (2 x) \cdot 2 - (1 + sin (x))

확대한다:

sin (x) cos (2 x) \cdot 2 - 1 - sin (x)

sin (x) cos (2 x) \cdot 2 - (1 + sin (x))

= 2 sin (x) cos (2 x) - (1 + sin (x))

- (1 + sin (x)) : - 1 - sin (x)

- (1 + sin (x))

괄호 배포

= - (1) - (sin (x))

마이너스 플러스 규칙 적용

+ (- a) = - a

= - 1 - sin (x)

= sin (x) cos (2 x) \cdot 2 - 1 - sin (x)

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

\frac{2 sin ( x ) cos ( 2 x ) - 1 - sin ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (x) cos (2 x) - 1 - sin (x) = 0

삼각성을 사용하여 다시 쓰기

- 1 - sin (x) + 2 cos (2 x) sin (x)

더블 앵글 아이덴티티 사용:

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

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

- 1 - sin (x) + 2 (1 - 2 sin^{2} (x)) sin (x)

간소화하다 :

- 1 + sin (x) - 4 sin^{3} (x)

- 1 - sin (x) + 2 (1 - 2 sin^{2} (x)) sin (x)

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

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

확대한다:

2 sin (x) - 4 sin^{3} (x)

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

분배 법칙 적용:

a (b - c) = ab - a c

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

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

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

2 \cdot 1 \cdot sin (x) - 2 \cdot 2 sin^{2} (x) sin (x)

단순화하세요:

2 sin (x) - 4 sin^{3} (x)

2 \cdot 1 \cdot sin (x) - 2 \cdot 2 sin^{2} (x) sin (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 \cdot sin (x)

숫자를 곱하시오:

2 \cdot 1 = 2

= 2 sin (x)

2 \cdot 2 sin^{2} (x) sin (x) = 4 sin^{3} (x)

2 \cdot 2 sin^{2} (x) sin (x)

숫자를 곱하시오:

2 \cdot 2 = 4

= 4 sin^{2} (x) sin (x)

지수 규칙 적용:

a^{b} \cdot a^{c} = a^{b + c}

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

= 4 sin^{2 + 1} (x)

숫자 추가:

2 + 1 = 3

= 4 sin^{3} (x)

= 2 sin (x) - 4 sin^{3} (x)

= 2 sin (x) - 4 sin^{3} (x)

= - 1 - sin (x) + 2 sin (x) - 4 sin^{3} (x)

유사 요소 추가:

- sin (x) + 2 sin (x) = sin (x)

= - 1 + sin (x) - 4 sin^{3} (x)

= - 1 + sin (x) - 4 sin^{3} (x)

- 1 + sin (x) - 4 sin^{3} (x) = 0

대체로 해결

- 1 + sin (x) - 4 sin^{3} (x) = 0

하게:

sin (x) = u

- 1 + u - 4 u^{3} = 0

- 1 + u - 4 u^{3} = 0 : u \approx - 0.76068 \dots

- 1 + u - 4 u^{3} = 0

표준 양식으로 작성

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 4 u^{3} + u - 1 = 0

다음을 위한 하나의 솔루션 찾기

- 4 u^{3} + u - 1 = 0

뉴턴-랩슨을 이용하여:

u \approx - 0.76068 \dots

- 4 u^{3} + u - 1 = 0

뉴턴-랩슨 근사 정의

f (u) = - 4 u^{3} + u - 1

f^{^{'}} (u)

찾다 :

- 12 u^{2} + 1

\frac{d}{d u} (- 4 u^{3} + u - 1)

합계/차이 규칙 적용:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (4 u^{3}) + \frac{d u}{d u} - \frac{d}{d u} (1)

\frac{d}{d u} (4 u^{3}) = 12 u^{2}

\frac{d}{d u} (4 u^{3})

정수를 빼라:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d u} (u^{3})

전원 규칙을 적용합니다:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 \cdot 3 u^{3 - 1}

단순화

= 12 u^{2}

\frac{d u}{d u} = 1

\frac{d u}{d u}

공통 도함수 적용:

\frac{d u}{d u} = 1

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

상수의 도함수:

\frac{d}{d x} (a) = 0

= 0

= - 12 u^{2} + 1 - 0

단순화

= - 12 u^{2} + 1

렛

u_{0} = - 1

계산하다

u_{n + 1}

까지

Δ u_{n + 1} < 0.000001

u_{1} = - 0.81818 \dots : Δ u_{1} = 0.18181 \dots

f (u_{0}) = - 4 (- 1)^{3} + (- 1) - 1 = 2

f^{^{'}} (u_{0}) = - 12 (- 1)^{2} + 1 = - 11

u_{1} = - 0.81818 \dots

Δ u_{1} = ∣ - 0.81818 \dots - (- 1) ∣ = 0.18181 \dots

Δ u_{1} = 0.18181 \dots

u_{2} = - 0.76519 \dots : Δ u_{2} = 0.05298 \dots

f (u_{1}) = - 4 (- 0.81818 \dots)^{3} + (- 0.81818 \dots) - 1 = 0.37265 \dots

f^{^{'}} (u_{1}) = - 12 (- 0.81818 \dots)^{2} + 1 = - 7.03305 \dots

u_{2} = - 0.76519 \dots

Δ u_{2} = ∣ - 0.76519 \dots - (- 0.81818 \dots) ∣ = 0.05298 \dots

Δ u_{2} = 0.05298 \dots

u_{3} = - 0.76072 \dots : Δ u_{3} = 0.00447 \dots

f (u_{2}) = - 4 (- 0.76519 \dots)^{3} + (- 0.76519 \dots) - 1 = 0.02696 \dots

f^{^{'}} (u_{2}) = - 12 (- 0.76519 \dots)^{2} + 1 = - 6.02629 \dots

u_{3} = - 0.76072 \dots

Δ u_{3} = ∣ - 0.76072 \dots - (- 0.76519 \dots) ∣ = 0.00447 \dots

Δ u_{3} = 0.00447 \dots

u_{4} = - 0.76068 \dots : Δ u_{4} = 0.00003 \dots

f (u_{3}) = - 4 (- 0.76072 \dots)^{3} + (- 0.76072 \dots) - 1 = 0.00018 \dots

f^{^{'}} (u_{3}) = - 12 (- 0.76072 \dots)^{2} + 1 = - 5.94435 \dots

u_{4} = - 0.76068 \dots

Δ u_{4} = ∣ - 0.76068 \dots - (- 0.76072 \dots) ∣ = 0.00003 \dots

Δ u_{4} = 0.00003 \dots

u_{5} = - 0.76068 \dots : Δ u_{5} = 1.46429 E - 9

f (u_{4}) = - 4 (- 0.76068 \dots)^{3} + (- 0.76068 \dots) - 1 = 8.70343 E - 9

f^{^{'}} (u_{4}) = - 12 (- 0.76068 \dots)^{2} + 1 = - 5.94378 \dots

u_{5} = - 0.76068 \dots

Δ u_{5} = ∣ - 0.76068 \dots - (- 0.76068 \dots) ∣ = 1.46429 E - 9

Δ u_{5} = 1.46429 E - 9

u \approx - 0.76068 \dots

긴 나눗셈 적용:

\frac{- 4 u ^{3} + u - 1}{u + 0.76068 \dots} = - 4 u^{2} + 3.04275 \dots u - 1.31459 \dots

- 4 u^{2} + 3.04275 \dots u - 1.31459 \dots \approx 0

다음을 위한 하나의 솔루션 찾기

- 4 u^{2} + 3.04275 \dots u - 1.31459 \dots = 0

뉴턴-랩슨을 이용하여:

솔루션 없음

u \in R

- 4 u^{2} + 3.04275 \dots u - 1.31459 \dots = 0

뉴턴-랩슨 근사 정의

f (u) = - 4 u^{2} + 3.04275 \dots u - 1.31459 \dots

f^{^{'}} (u)

찾다 :

- 8 u + 3.04275 \dots

\frac{d}{d u} (- 4 u^{2} + 3.04275 \dots u - 1.31459 \dots)

합계/차이 규칙 적용:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d u} (4 u^{2}) + \frac{d}{d u} (3.04275 \dots u) - \frac{d}{d u} (1.31459 \dots)

\frac{d}{d u} (4 u^{2}) = 8 u

\frac{d}{d u} (4 u^{2})

정수를 빼라:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d u} (u^{2})

전원 규칙을 적용합니다:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 \cdot 2 u^{2 - 1}

단순화

= 8 u

\frac{d}{d u} (3.04275 \dots u) = 3.04275 \dots

\frac{d}{d u} (3.04275 \dots u)

정수를 빼라:

(a \cdot f)^{'} = a \cdot f^{'}

= 3.04275 \dots \frac{d u}{d u}

공통 도함수 적용:

\frac{d u}{d u} = 1

= 3.04275 \dots \cdot 1

단순화

= 3.04275 \dots

\frac{d}{d u} (1.31459 \dots) = 0

\frac{d}{d u} (1.31459 \dots)

상수의 도함수:

\frac{d}{d x} (a) = 0

= 0

= - 8 u + 3.04275 \dots - 0

단순화

= - 8 u + 3.04275 \dots

렛

u_{0} = 0

계산하다

u_{n + 1}

까지

Δ u_{n + 1} < 0.000001

u_{1} = 0.43204 \dots : Δ u_{1} = 0.43204 \dots

f (u_{0}) = - 4 \cdot 0^{2} + 3.04275 \dots \cdot 0 - 1.31459 \dots = - 1.31459 \dots

f^{^{'}} (u_{0}) = - 8 \cdot 0 + 3.04275 \dots = 3.04275 \dots

u_{1} = 0.43204 \dots

Δ u_{1} = ∣ 0.43204 \dots - 0 ∣ = 0.43204 \dots

Δ u_{1} = 0.43204 \dots

u_{2} = - 1.37331 \dots : Δ u_{2} = 1.80535 \dots

f (u_{1}) = - 4 \cdot 0.43204 \dots^{2} + 3.04275 \dots \cdot 0.43204 \dots - 1.31459 \dots = - 0.74663 \dots

f^{^{'}} (u_{1}) = - 8 \cdot 0.43204 \dots + 3.04275 \dots = - 0.41356 \dots

u_{2} = - 1.37331 \dots

Δ u_{2} = ∣ - 1.37331 \dots - 0.43204 \dots ∣ = 1.80535 \dots

Δ u_{2} = 1.80535 \dots

u_{3} = - 0.44402 \dots : Δ u_{3} = 0.92928 \dots

f (u_{2}) = - 4 (- 1.37331 \dots)^{2} + 3.04275 \dots (- 1.37331 \dots) - 1.31459 \dots = - 13.03728 \dots

f^{^{'}} (u_{2}) = - 8 (- 1.37331 \dots) + 3.04275 \dots = 14.02930 \dots

u_{3} = - 0.44402 \dots

Δ u_{3} = ∣ - 0.44402 \dots - (- 1.37331 \dots) ∣ = 0.92928 \dots

Δ u_{3} = 0.92928 \dots

u_{4} = 0.07974 \dots : Δ u_{4} = 0.52377 \dots

f (u_{3}) = - 4 (- 0.44402 \dots)^{2} + 3.04275 \dots (- 0.44402 \dots) - 1.31459 \dots = - 3.45431 \dots

f^{^{'}} (u_{3}) = - 8 (- 0.44402 \dots) + 3.04275 \dots = 6.59499 \dots

u_{4} = 0.07974 \dots

Δ u_{4} = ∣ 0.07974 \dots - (- 0.44402 \dots) ∣ = 0.52377 \dots

Δ u_{4} = 0.52377 \dots

u_{5} = 0.53608 \dots : Δ u_{5} = 0.45633 \dots

f (u_{4}) = - 4 \cdot 0.07974 \dots^{2} + 3.04275 \dots \cdot 0.07974 \dots - 1.31459 \dots = - 1.09737 \dots

f^{^{'}} (u_{4}) = - 8 \cdot 0.07974 \dots + 3.04275 \dots = 2.40476 \dots

u_{5} = 0.53608 \dots

Δ u_{5} = ∣ 0.53608 \dots - 0.07974 \dots ∣ = 0.45633 \dots

Δ u_{5} = 0.45633 \dots

u_{6} = - 0.13247 \dots : Δ u_{6} = 0.66855 \dots

f (u_{5}) = - 4 \cdot 0.53608 \dots^{2} + 3.04275 \dots \cdot 0.53608 \dots - 1.31459 \dots = - 0.83296 \dots

f^{^{'}} (u_{5}) = - 8 \cdot 0.53608 \dots + 3.04275 \dots = - 1.24592 \dots

u_{6} = - 0.13247 \dots

Δ u_{6} = ∣ - 0.13247 \dots - 0.53608 \dots ∣ = 0.66855 \dots

Δ u_{6} = 0.66855 \dots

u_{7} = 0.30332 \dots : Δ u_{7} = 0.43579 \dots

f (u_{6}) = - 4 (- 0.13247 \dots)^{2} + 3.04275 \dots (- 0.13247 \dots) - 1.31459 \dots = - 1.78786 \dots

f^{^{'}} (u_{6}) = - 8 (- 0.13247 \dots) + 3.04275 \dots = 4.10252 \dots

u_{7} = 0.30332 \dots

Δ u_{7} = ∣ 0.30332 \dots - (- 0.13247 \dots) ∣ = 0.43579 \dots

Δ u_{7} = 0.43579 \dots

u_{8} = 1.53626 \dots : Δ u_{8} = 1.23293 \dots

f (u_{7}) = - 4 \cdot 0.30332 \dots^{2} + 3.04275 \dots \cdot 0.30332 \dots - 1.31459 \dots = - 0.75967 \dots

f^{^{'}} (u_{7}) = - 8 \cdot 0.30332 \dots + 3.04275 \dots = 0.61615 \dots

u_{8} = 1.53626 \dots

Δ u_{8} = ∣ 1.53626 \dots - 0.30332 \dots ∣ = 1.23293 \dots

Δ u_{8} = 1.23293 \dots

u_{9} = 0.87871 \dots : Δ u_{9} = 0.65754 \dots

f (u_{8}) = - 4 \cdot 1.53626 \dots^{2} + 3.04275 \dots \cdot 1.53626 \dots - 1.31459 \dots = - 6.08050 \dots

f^{^{'}} (u_{8}) = - 8 \cdot 1.53626 \dots + 3.04275 \dots = - 9.24732 \dots

u_{9} = 0.87871 \dots

Δ u_{9} = ∣ 0.87871 \dots - 1.53626 \dots ∣ = 0.65754 \dots

Δ u_{9} = 0.65754 \dots

u_{10} = 0.44494 \dots : Δ u_{10} = 0.43377 \dots

f (u_{9}) = - 4 \cdot 0.87871 \dots^{2} + 3.04275 \dots \cdot 0.87871 \dots - 1.31459 \dots = - 1.72944 \dots

f^{^{'}} (u_{9}) = - 8 \cdot 0.87871 \dots + 3.04275 \dots = - 3.98698 \dots

u_{10} = 0.44494 \dots

Δ u_{10} = ∣ 0.44494 \dots - 0.87871 \dots ∣ = 0.43377 \dots

Δ u_{10} = 0.43377 \dots

u_{11} = - 1.01142 \dots : Δ u_{11} = 1.45637 \dots

f (u_{10}) = - 4 \cdot 0.44494 \dots^{2} + 3.04275 \dots \cdot 0.44494 \dots - 1.31459 \dots = - 0.75263 \dots

f^{^{'}} (u_{10}) = - 8 \cdot 0.44494 \dots + 3.04275 \dots = - 0.51679 \dots

u_{11} = - 1.01142 \dots

Δ u_{11} = ∣ - 1.01142 \dots - 0.44494 \dots ∣ = 1.45637 \dots

Δ u_{11} = 1.45637 \dots

해결 방법을 찾을 수 없습니다

해결책은

u \approx - 0.76068 \dots

뒤로 대체

u = sin (x)

sin (x) \approx - 0.76068 \dots

sin (x) \approx - 0.76068 \dots

sin (x) = - 0.76068 \dots : x = arcsin (- 0.76068 \dots) + 2 πn, x = π + arcsin (0.76068 \dots) + 2 πn

sin (x) = - 0.76068 \dots

트리거 역속성 적용

sin (x) = - 0.76068 \dots

일반 솔루션

sin (x) = - 0.76068 \dots

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- 0.76068 \dots) + 2 πn, x = π + arcsin (0.76068 \dots) + 2 πn

x = arcsin (- 0.76068 \dots) + 2 πn, x = π + arcsin (0.76068 \dots) + 2 πn

모든 솔루션 결합

x = arcsin (- 0.76068 \dots) + 2 πn, x = π + arcsin (0.76068 \dots) + 2 πn

해를 10진수 형식으로 표시

x = - 0.86437 \dots + 2 πn, x = π + 0.86437 \dots + 2 πn

그래프

인기 있는 예

e^{s i n (x)} = 2

(sec(x))/(cot(x)tan(x))=sin(x)

\frac{sec ( x )}{cot ( x ) tan ( x )} = sin (x)

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

4sin(θ)+2sqrt(3)=0

4 sin (θ) + 23 = 0

400+290cos(x)+290sin(x)=0

400 + 290 cos (x) + 290 sin (x) = 0