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人気のある >

(0.9)/(cos(x))+(0.7*sin(x))=1.1

解

\frac{0.9}{cos ( x )} + (0.7 \cdot sin (x)) = 1.1

解

x = 0.24774 \dots + 2 πn, x = - 1.01026 \dots + 2 πn

+1

度

x = 14.19491 \dots^{\circ} + 36 0^{\circ} n, x = - 57.88377 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

\frac{0.9}{cos ( x )} + (0.7 sin (x)) = 1.1

両辺から

0.7 sin (x)

を引く

\frac{0.9}{cos ( x )} = 1.1 - 0.7 sin (x)

両辺を2乗する

(\frac{0.9}{cos ( x )})^{2} = (1.1 - 0.7 sin (x))^{2}

両辺から

(1.1 - 0.7 sin (x))^{2}

を引く

\frac{0.81}{cos ^{2} ( x )} - 1.21 + 1.54 sin (x) - 0.49 sin^{2} (x) = 0

三角関数の公式を使用して書き換える

- 1.21 + \frac{0.81}{cos ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x)

ピタゴラスの公式を使用する:

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

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

= - 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x)

- 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x) = 0

置換で解く

- 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x) = 0

仮定:

sin (x) = u

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0 : u \approx 0.24522 \dots, u \approx - 0.84697 \dots

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0

以下で両辺を乗じる:

100

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0

小数点を取り除くには, 小数点以下の各桁に10を乗じます小数点の右側は

2

桁なので,

100 を乗じます

- 1.21 \cdot 100 + \frac{0.81}{1 - u ^{2}} \cdot 100 - 0.49 u^{2} \cdot 100 + 1.54 u \cdot 100 = 0 \cdot 100

改良

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

以下で両辺を乗じる:

1 - u^{2}

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

以下で両辺を乗じる:

1 - u^{2}

- 121 (1 - u^{2}) + \frac{81}{1 - u ^{2}} (1 - u^{2}) - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 \cdot (1 - u^{2})

簡素化

- 121 (1 - u^{2}) + \frac{81}{1 - u ^{2}} (1 - u^{2}) - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 \cdot (1 - u^{2})

簡素化

\frac{81}{1 - u ^{2}} (1 - u^{2}) : 81

\frac{81}{1 - u ^{2}} (1 - u^{2})

分数を乗じる:

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

= \frac{81 ( 1 - u ^{2} )}{1 - u ^{2}}

共通因数を約分する:

1 - u^{2}

= 81

簡素化

0 \cdot (1 - u^{2}) : 0

0 \cdot (1 - u^{2})

規則を適用

0 \cdot a = 0

= 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

解く

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 : u \approx 0.24522 \dots, u \approx - 0.84697 \dots

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

拡張

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) : 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2})

拡張

- 121 (1 - u^{2}) : - 121 + 121 u^{2}

- 121 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = - 121, b = 1, c = u^{2}

= - 121 \cdot 1 - (- 121) u^{2}

マイナス・プラスの規則を適用する

- (- a) = a

= - 121 \cdot 1 + 121 u^{2}

数を乗じる:

121 \cdot 1 = 121

= - 121 + 121 u^{2}

= - 121 + 121 u^{2} + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2})

拡張

- 49 u^{2} (1 - u^{2}) : - 49 u^{2} + 49 u^{4}

- 49 u^{2} (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = - 49 u^{2}, b = 1, c = u^{2}

= - 49 u^{2} \cdot 1 - (- 49 u^{2}) u^{2}

マイナス・プラスの規則を適用する

- (- a) = a

= - 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2}

簡素化

- 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2} : - 49 u^{2} + 49 u^{4}

- 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2}

49 \cdot 1 \cdot u^{2} = 49 u^{2}

49 \cdot 1 \cdot u^{2}

数を乗じる:

49 \cdot 1 = 49

= 49 u^{2}

49 u^{2} u^{2} = 49 u^{4}

49 u^{2} u^{2}

指数の規則を適用する:

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

u^{2} u^{2} = u^{2 + 2}

= 49 u^{2 + 2}

数を足す:

2 + 2 = 4

= 49 u^{4}

= - 49 u^{2} + 49 u^{4}

= - 49 u^{2} + 49 u^{4}

= - 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u (1 - u^{2})

拡張

154 u (1 - u^{2}) : 154 u - 154 u^{3}

154 u (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 154 u, b = 1, c = u^{2}

= 154 u \cdot 1 - 154 u u^{2}

= 154 \cdot 1 \cdot u - 154 u^{2} u

簡素化

154 \cdot 1 \cdot u - 154 u^{2} u : 154 u - 154 u^{3}

154 \cdot 1 \cdot u - 154 u^{2} u

154 \cdot 1 \cdot u = 154 u

154 \cdot 1 \cdot u

数を乗じる:

154 \cdot 1 = 154

= 154 u

154 u^{2} u = 154 u^{3}

154 u^{2} u

指数の規則を適用する:

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

u^{2} u = u^{2 + 1}

= 154 u^{2 + 1}

数を足す:

2 + 1 = 3

= 154 u^{3}

= 154 u - 154 u^{3}

= 154 u - 154 u^{3}

= - 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3}

簡素化

- 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3} : 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

- 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3}

条件のようなグループ

= 49 u^{4} - 154 u^{3} + 121 u^{2} - 49 u^{2} + 154 u - 121 + 81

類似した元を足す:

121 u^{2} - 49 u^{2} = 72 u^{2}

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 121 + 81

数を足す/引く:

- 121 + 81 = - 40

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

ニュートン・ラプソン法を使用して

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

の解を1つ求める:

u \approx 0.24522 \dots

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

ニュートン・ラプソン概算の定義

f (u) = 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

発見する

f^{^{'}} (u) : 196 u^{3} - 462 u^{2} + 144 u + 154

\frac{d}{d u} (49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40)

和/差の法則を適用:

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

= \frac{d}{d u} (49 u^{4}) - \frac{d}{d u} (154 u^{3}) + \frac{d}{d u} (72 u^{2}) + \frac{d}{d u} (154 u) - \frac{d}{d u} (40)

\frac{d}{d u} (49 u^{4}) = 196 u^{3}

\frac{d}{d u} (49 u^{4})

定数を除去:

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

= 49 \frac{d}{d u} (u^{4})

乗の法則を適用:

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

= 49 \cdot 4 u^{4 - 1}

簡素化

= 196 u^{3}

\frac{d}{d u} (154 u^{3}) = 462 u^{2}

\frac{d}{d u} (154 u^{3})

定数を除去:

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

= 154 \frac{d}{d u} (u^{3})

乗の法則を適用:

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

= 154 \cdot 3 u^{3 - 1}

簡素化

= 462 u^{2}

\frac{d}{d u} (72 u^{2}) = 144 u

\frac{d}{d u} (72 u^{2})

定数を除去:

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

= 72 \frac{d}{d u} (u^{2})

乗の法則を適用:

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

= 72 \cdot 2 u^{2 - 1}

簡素化

= 144 u

\frac{d}{d u} (154 u) = 154

\frac{d}{d u} (154 u)

定数を除去:

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

= 154 \frac{d u}{d u}

共通の導関数を適用:

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

= 154 \cdot 1

簡素化

= 154

\frac{d}{d u} (40) = 0

\frac{d}{d u} (40)

定数の導関数:

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

= 0

= 196 u^{3} - 462 u^{2} + 144 u + 154 - 0

簡素化

= 196 u^{3} - 462 u^{2} + 144 u + 154

仮定:

u_{0} = 0

Δ u_{n + 1} <

になるまで

u_{n + 1}

を計算する

0.000001

u_{1} = 0.25974 \dots : Δ u_{1} = 0.25974 \dots

f (u_{0}) = 49 \cdot 0^{4} - 154 \cdot 0^{3} + 72 \cdot 0^{2} + 154 \cdot 0 - 40 = - 40

f^{^{'}} (u_{0}) = 196 \cdot 0^{3} - 462 \cdot 0^{2} + 144 \cdot 0 + 154 = 154

u_{1} = 0.25974 \dots

Δ u_{1} = ∣ 0.25974 \dots - 0 ∣ = 0.25974 \dots

Δ u_{1} = 0.25974 \dots

u_{2} = 0.24518 \dots : Δ u_{2} = 0.01455 \dots

f (u_{1}) = 49 \cdot 0.25974 \dots^{4} - 154 \cdot 0.25974 \dots^{3} + 72 \cdot 0.25974 \dots^{2} + 154 \cdot 0.25974 \dots - 40 = 2.38190 \dots

f^{^{'}} (u_{1}) = 196 \cdot 0.25974 \dots^{3} - 462 \cdot 0.25974 \dots^{2} + 144 \cdot 0.25974 \dots + 154 = 163.66834 \dots

u_{2} = 0.24518 \dots

Δ u_{2} = ∣ 0.24518 \dots - 0.25974 \dots ∣ = 0.01455 \dots

Δ u_{2} = 0.01455 \dots

u_{3} = 0.24522 \dots : Δ u_{3} = 0.00003 \dots

f (u_{2}) = 49 \cdot 0.24518 \dots^{4} - 154 \cdot 0.24518 \dots^{3} + 72 \cdot 0.24518 \dots^{2} + 154 \cdot 0.24518 \dots - 40 = - 0.00564 \dots

f^{^{'}} (u_{2}) = 196 \cdot 0.24518 \dots^{3} - 462 \cdot 0.24518 \dots^{2} + 144 \cdot 0.24518 \dots + 154 = 164.42203 \dots

u_{3} = 0.24522 \dots

Δ u_{3} = ∣ 0.24522 \dots - 0.24518 \dots ∣ = 0.00003 \dots

Δ u_{3} = 0.00003 \dots

u_{4} = 0.24522 \dots : Δ u_{4} = 1.69249 E - 10

f (u_{3}) = 49 \cdot 0.24522 \dots^{4} - 154 \cdot 0.24522 \dots^{3} + 72 \cdot 0.24522 \dots^{2} + 154 \cdot 0.24522 \dots - 40 = - 2.78279 E - 8

f^{^{'}} (u_{3}) = 196 \cdot 0.24522 \dots^{3} - 462 \cdot 0.24522 \dots^{2} + 144 \cdot 0.24522 \dots + 154 = 164.42041 \dots

u_{4} = 0.24522 \dots

Δ u_{4} = ∣ 0.24522 \dots - 0.24522 \dots ∣ = 1.69249 E - 10

Δ u_{4} = 1.69249 E - 10

u \approx 0.24522 \dots

長除法を適用する:

\frac{49 u ^{4} - 154 u ^{3} + 72 u ^{2} + 154 u - 40}{u - 0.24522 \dots} = 49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots \approx 0

ニュートン・ラプソン法を使用して

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots = 0

の解を1つ求める:

u \approx - 0.84697 \dots

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots = 0

ニュートン・ラプソン概算の定義

f (u) = 49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots

発見する

f^{^{'}} (u) : 147 u^{2} - 283.96830 \dots u + 37.18245 \dots

\frac{d}{d u} (49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots)

和/差の法則を適用:

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

= \frac{d}{d u} (49 u^{3}) - \frac{d}{d u} (141.98415 \dots u^{2}) + \frac{d}{d u} (37.18245 \dots u) + \frac{d}{d u} (163.11793 \dots)

\frac{d}{d u} (49 u^{3}) = 147 u^{2}

\frac{d}{d u} (49 u^{3})

定数を除去:

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

= 49 \frac{d}{d u} (u^{3})

乗の法則を適用:

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

= 49 \cdot 3 u^{3 - 1}

簡素化

= 147 u^{2}

\frac{d}{d u} (141.98415 \dots u^{2}) = 283.96830 \dots u

\frac{d}{d u} (141.98415 \dots u^{2})

定数を除去:

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

= 141.98415 \dots \frac{d}{d u} (u^{2})

乗の法則を適用:

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

= 141.98415 \dots \cdot 2 u^{2 - 1}

簡素化

= 283.96830 \dots u

\frac{d}{d u} (37.18245 \dots u) = 37.18245 \dots

\frac{d}{d u} (37.18245 \dots u)

定数を除去:

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

= 37.18245 \dots \frac{d u}{d u}

共通の導関数を適用:

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

= 37.18245 \dots \cdot 1

簡素化

= 37.18245 \dots

\frac{d}{d u} (163.11793 \dots) = 0

\frac{d}{d u} (163.11793 \dots)

定数の導関数:

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

= 0

= 147 u^{2} - 283.96830 \dots u + 37.18245 \dots + 0

簡素化

= 147 u^{2} - 283.96830 \dots u + 37.18245 \dots

仮定:

u_{0} = - 4

Δ u_{n + 1} <

になるまで

u_{n + 1}

を計算する

0.000001

u_{1} = - 2.46999 \dots : Δ u_{1} = 1.53000 \dots

f (u_{0}) = 49 (- 4)^{3} - 141.98415 \dots (- 4)^{2} + 37.18245 \dots (- 4) + 163.11793 \dots = - 5393.35834 \dots

f^{^{'}} (u_{0}) = 147 (- 4)^{2} - 283.96830 \dots (- 4) + 37.18245 \dots = 3525.05568 \dots

u_{1} = - 2.46999 \dots

Δ u_{1} = ∣ - 2.46999 \dots - (- 4) ∣ = 1.53000 \dots

Δ u_{1} = 1.53000 \dots

u_{2} = - 1.53240 \dots : Δ u_{2} = 0.93758 \dots

f (u_{1}) = 49 (- 2.46999 \dots)^{3} - 141.98415 \dots (- 2.46999 \dots)^{2} + 37.18245 \dots (- 2.46999 \dots) + 163.11793 \dots = - 1533.33504 \dots

f^{^{'}} (u_{1}) = 147 (- 2.46999 \dots)^{2} - 283.96830 \dots (- 2.46999 \dots) + 37.18245 \dots = 1635.40983 \dots

u_{2} = - 1.53240 \dots

Δ u_{2} = ∣ - 1.53240 \dots - (- 2.46999 \dots) ∣ = 0.93758 \dots

Δ u_{2} = 0.93758 \dots

u_{3} = - 1.03872 \dots : Δ u_{3} = 0.49368 \dots

f (u_{2}) = 49 (- 1.53240 \dots)^{3} - 141.98415 \dots (- 1.53240 \dots)^{2} + 37.18245 \dots (- 1.53240 \dots) + 163.11793 \dots = - 403.60639 \dots

f^{^{'}} (u_{2}) = 147 (- 1.53240 \dots)^{2} - 283.96830 \dots (- 1.53240 \dots) + 37.18245 \dots = 817.53470 \dots

u_{3} = - 1.03872 \dots

Δ u_{3} = ∣ - 1.03872 \dots - (- 1.53240 \dots) ∣ = 0.49368 \dots

Δ u_{3} = 0.49368 \dots

u_{4} = - 0.86834 \dots : Δ u_{4} = 0.17037 \dots

f (u_{3}) = 49 (- 1.03872 \dots)^{3} - 141.98415 \dots (- 1.03872 \dots)^{2} + 37.18245 \dots (- 1.03872 \dots) + 163.11793 \dots = - 83.61240 \dots

f^{^{'}} (u_{3}) = 147 (- 1.03872 \dots)^{2} - 283.96830 \dots (- 1.03872 \dots) + 37.18245 \dots = 490.75107 \dots

u_{4} = - 0.86834 \dots

Δ u_{4} = ∣ - 0.86834 \dots - (- 1.03872 \dots) ∣ = 0.17037 \dots

Δ u_{4} = 0.17037 \dots

u_{5} = - 0.84728 \dots : Δ u_{5} = 0.02106 \dots

f (u_{4}) = 49 (- 0.86834 \dots)^{3} - 141.98415 \dots (- 0.86834 \dots)^{2} + 37.18245 \dots (- 0.86834 \dots) + 163.11793 \dots = - 8.31155 \dots

f^{^{'}} (u_{4}) = 147 (- 0.86834 \dots)^{2} - 283.96830 \dots (- 0.86834 \dots) + 37.18245 \dots = 394.60645 \dots

u_{5} = - 0.84728 \dots

Δ u_{5} = ∣ - 0.84728 \dots - (- 0.86834 \dots) ∣ = 0.02106 \dots

Δ u_{5} = 0.02106 \dots

u_{6} = - 0.84697 \dots : Δ u_{6} = 0.00031 \dots

f (u_{5}) = 49 (- 0.84728 \dots)^{3} - 141.98415 \dots (- 0.84728 \dots)^{2} + 37.18245 \dots (- 0.84728 \dots) + 163.11793 \dots = - 0.11916 \dots

f^{^{'}} (u_{5}) = 147 (- 0.84728 \dots)^{2} - 283.96830 \dots (- 0.84728 \dots) + 37.18245 \dots = 383.31325 \dots

u_{6} = - 0.84697 \dots

Δ u_{6} = ∣ - 0.84697 \dots - (- 0.84728 \dots) ∣ = 0.00031 \dots

Δ u_{6} = 0.00031 \dots

u_{7} = - 0.84697 \dots : Δ u_{7} = 6.72258 E - 8

f (u_{6}) = 49 (- 0.84697 \dots)^{3} - 141.98415 \dots (- 0.84697 \dots)^{2} + 37.18245 \dots (- 0.84697 \dots) + 163.11793 \dots = - 0.00002 \dots

f^{^{'}} (u_{6}) = 147 (- 0.84697 \dots)^{2} - 283.96830 \dots (- 0.84697 \dots) + 37.18245 \dots = 383.14754 \dots

u_{7} = - 0.84697 \dots

Δ u_{7} = ∣ - 0.84697 \dots - (- 0.84697 \dots) ∣ = 6.72258 E - 8

Δ u_{7} = 6.72258 E - 8

u \approx - 0.84697 \dots

長除法を適用する:

\frac{49 u ^{3} - 141.98415 \dots u ^{2} + 37.18245 \dots u + 163.11793 \dots}{u + 0.84697 \dots} = 49 u^{2} - 183.48575 \dots u + 192.58964 \dots

49 u^{2} - 183.48575 \dots u + 192.58964 \dots \approx 0

ニュートン・ラプソン法を使用して

49 u^{2} - 183.48575 \dots u + 192.58964 \dots = 0

の解を1つ求める:

以下の解はない:

u \in R

49 u^{2} - 183.48575 \dots u + 192.58964 \dots = 0

ニュートン・ラプソン概算の定義

f (u) = 49 u^{2} - 183.48575 \dots u + 192.58964 \dots

発見する

f^{^{'}} (u) : 98 u - 183.48575 \dots

\frac{d}{d u} (49 u^{2} - 183.48575 \dots u + 192.58964 \dots)

和/差の法則を適用:

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

= \frac{d}{d u} (49 u^{2}) - \frac{d}{d u} (183.48575 \dots u) + \frac{d}{d u} (192.58964 \dots)

\frac{d}{d u} (49 u^{2}) = 98 u

\frac{d}{d u} (49 u^{2})

定数を除去:

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

= 49 \frac{d}{d u} (u^{2})

乗の法則を適用:

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

= 49 \cdot 2 u^{2 - 1}

簡素化

= 98 u

\frac{d}{d u} (183.48575 \dots u) = 183.48575 \dots

\frac{d}{d u} (183.48575 \dots u)

定数を除去:

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

= 183.48575 \dots \frac{d u}{d u}

共通の導関数を適用:

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

= 183.48575 \dots \cdot 1

簡素化

= 183.48575 \dots

\frac{d}{d u} (192.58964 \dots) = 0

\frac{d}{d u} (192.58964 \dots)

定数の導関数:

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

= 0

= 98 u - 183.48575 \dots + 0

簡素化

= 98 u - 183.48575 \dots

仮定:

u_{0} = 1

Δ u_{n + 1} <

になるまで

u_{n + 1}

を計算する

0.000001

u_{1} = 1.67969 \dots : Δ u_{1} = 0.67969 \dots

f (u_{0}) = 49 \cdot 1^{2} - 183.48575 \dots \cdot 1 + 192.58964 \dots = 58.10389 \dots

f^{^{'}} (u_{0}) = 98 \cdot 1 - 183.48575 \dots = - 85.48575 \dots

u_{1} = 1.67969 \dots

Δ u_{1} = ∣ 1.67969 \dots - 1 ∣ = 0.67969 \dots

Δ u_{1} = 0.67969 \dots

u_{2} = 2.87893 \dots : Δ u_{2} = 1.19924 \dots

f (u_{1}) = 49 \cdot 1.67969 \dots^{2} - 183.48575 \dots \cdot 1.67969 \dots + 192.58964 \dots = 22.63700 \dots

f^{^{'}} (u_{1}) = 98 \cdot 1.67969 \dots - 183.48575 \dots = - 18.87604 \dots

u_{2} = 2.87893 \dots

Δ u_{2} = ∣ 2.87893 \dots - 1.67969 \dots ∣ = 1.19924 \dots

Δ u_{2} = 1.19924 \dots

u_{3} = 2.16457 \dots : Δ u_{3} = 0.71435 \dots

f (u_{2}) = 49 \cdot 2.87893 \dots^{2} - 183.48575 \dots \cdot 2.87893 \dots + 192.58964 \dots = 70.47130 \dots

f^{^{'}} (u_{2}) = 98 \cdot 2.87893 \dots - 183.48575 \dots = 98.65002 \dots

u_{3} = 2.16457 \dots

Δ u_{3} = ∣ 2.16457 \dots - 2.87893 \dots ∣ = 0.71435 \dots

Δ u_{3} = 0.71435 \dots

u_{4} = 1.29159 \dots : Δ u_{4} = 0.87298 \dots

f (u_{3}) = 49 \cdot 2.16457 \dots^{2} - 183.48575 \dots \cdot 2.16457 \dots + 192.58964 \dots = 25.00497 \dots

f^{^{'}} (u_{3}) = 98 \cdot 2.16457 \dots - 183.48575 \dots = 28.64306 \dots

u_{4} = 1.29159 \dots

Δ u_{4} = ∣ 1.29159 \dots - 2.16457 \dots ∣ = 0.87298 \dots

Δ u_{4} = 0.87298 \dots

u_{5} = 1.94777 \dots : Δ u_{5} = 0.65618 \dots

f (u_{4}) = 49 \cdot 1.29159 \dots^{2} - 183.48575 \dots \cdot 1.29159 \dots + 192.58964 \dots = 37.34304 \dots

f^{^{'}} (u_{4}) = 98 \cdot 1.29159 \dots - 183.48575 \dots = - 56.90947 \dots

u_{5} = 1.94777 \dots

Δ u_{5} = ∣ 1.94777 \dots - 1.29159 \dots ∣ = 0.65618 \dots

Δ u_{5} = 0.65618 \dots

u_{6} = - 0.90468 \dots : Δ u_{6} = 2.85246 \dots

f (u_{5}) = 49 \cdot 1.94777 \dots^{2} - 183.48575 \dots \cdot 1.94777 \dots + 192.58964 \dots = 21.09824 \dots

f^{^{'}} (u_{5}) = 98 \cdot 1.94777 \dots - 183.48575 \dots = 7.39649 \dots

u_{6} = - 0.90468 \dots

Δ u_{6} = ∣ - 0.90468 \dots - 1.94777 \dots ∣ = 2.85246 \dots

Δ u_{6} = 2.85246 \dots

u_{7} = 0.56030 \dots : Δ u_{7} = 1.46499 \dots

f (u_{6}) = 49 (- 0.90468 \dots)^{2} - 183.48575 \dots (- 0.90468 \dots) + 192.58964 \dots = 398.69183 \dots

f^{^{'}} (u_{6}) = 98 (- 0.90468 \dots) - 183.48575 \dots = - 272.14527 \dots

u_{7} = 0.56030 \dots

Δ u_{7} = ∣ 0.56030 \dots - (- 0.90468 \dots) ∣ = 1.46499 \dots

Δ u_{7} = 1.46499 \dots

u_{8} = 1.37822 \dots : Δ u_{8} = 0.81791 \dots

f (u_{7}) = 49 \cdot 0.56030 \dots^{2} - 183.48575 \dots \cdot 0.56030 \dots + 192.58964 \dots = 105.16450 \dots

f^{^{'}} (u_{7}) = 98 \cdot 0.56030 \dots - 183.48575 \dots = - 128.57563 \dots

u_{8} = 1.37822 \dots

Δ u_{8} = ∣ 1.37822 \dots - 0.56030 \dots ∣ = 0.81791 \dots

Δ u_{8} = 0.81791 \dots

u_{9} = 2.05523 \dots : Δ u_{9} = 0.67701 \dots

f (u_{8}) = 49 \cdot 1.37822 \dots^{2} - 183.48575 \dots \cdot 1.37822 \dots + 192.58964 \dots = 32.78061 \dots

f^{^{'}} (u_{8}) = 98 \cdot 1.37822 \dots - 183.48575 \dots = - 48.41953 \dots

u_{9} = 2.05523 \dots

Δ u_{9} = ∣ 2.05523 \dots - 1.37822 \dots ∣ = 0.67701 \dots

Δ u_{9} = 0.67701 \dots

u_{10} = 0.80248 \dots : Δ u_{10} = 1.25275 \dots

f (u_{9}) = 49 \cdot 2.05523 \dots^{2} - 183.48575 \dots \cdot 2.05523 \dots + 192.58964 \dots = 22.45892 \dots

f^{^{'}} (u_{9}) = 98 \cdot 2.05523 \dots - 183.48575 \dots = 17.92765 \dots

u_{10} = 0.80248 \dots

Δ u_{10} = ∣ 0.80248 \dots - 2.05523 \dots ∣ = 1.25275 \dots

Δ u_{10} = 1.25275 \dots

解を見つけられない

解答は

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

解を検算する

未定義の (特異) 点を求める:

u = 1, u = - 1

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u

の分母をゼロに比較する

解く

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

1

を右側に移動します

1 - u^{2} = 0

両辺から

1

を引く

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

簡素化

- u^{2} = - 1

- u^{2} = - 1

以下で両辺を割る

- 1

- u^{2} = - 1

以下で両辺を割る

- 1

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

簡素化

u^{2} = 1

u^{2} = 1

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

累乗根の規則を適用する:

1 = 1

= 1

- 1 = - 1

- 1

累乗根の規則を適用する:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

以下の点は定義されていない

u = 1, u = - 1

未定義のポイントを解に組み合わせる:

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

代用を戻す

u = sin (x)

sin (x) \approx 0.24522 \dots, sin (x) \approx - 0.84697 \dots

sin (x) \approx 0.24522 \dots, sin (x) \approx - 0.84697 \dots

sin (x) = 0.24522 \dots : x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

sin (x) = 0.24522 \dots

三角関数の逆数プロパティを適用する

sin (x) = 0.24522 \dots

以下の一般解

sin (x) = 0.24522 \dots

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

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

sin (x) = - 0.84697 \dots : x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

sin (x) = - 0.84697 \dots

三角関数の逆数プロパティを適用する

sin (x) = - 0.84697 \dots

以下の一般解

sin (x) = - 0.84697 \dots

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

x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

すべての解を組み合わせる

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn, x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

元のequationに当てはめて解を検算する

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

arcsin (0.24522 \dots) + 2 πn :

真

arcsin (0.24522 \dots) + 2 πn

挿入

n = 1

arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

の挿入向け

x = arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( arcsin ( 0.24522 \dots ) + 2 π 1 )} + 0.7 sin (arcsin (0.24522 \dots) + 2 π 1) = 1.1

改良

1.1 = 1.1

\Rightarrow 真

解答を確認する

π - arcsin (0.24522 \dots) + 2 πn :

偽

π - arcsin (0.24522 \dots) + 2 πn

挿入

n = 1

π - arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

の挿入向け

x = π - arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( π - arcsin ( 0.24522 \dots ) + 2 π 1 )} + 0.7 sin (π - arcsin (0.24522 \dots) + 2 π 1) = 1.1

改良

- 0.75669 \dots = 1.1

\Rightarrow 偽

解答を確認する

arcsin (- 0.84697 \dots) + 2 πn :

真

arcsin (- 0.84697 \dots) + 2 πn

挿入

n = 1

arcsin (- 0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

の挿入向け

x = arcsin (- 0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( arcsin ( - 0.84697 \dots ) + 2 π 1 )} + 0.7 sin (arcsin (- 0.84697 \dots) + 2 π 1) = 1.1

改良

1.1 = 1.1

\Rightarrow 真

解答を確認する

π + arcsin (0.84697 \dots) + 2 πn :

偽

π + arcsin (0.84697 \dots) + 2 πn

挿入

n = 1

π + arcsin (0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

の挿入向け

x = π + arcsin (0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( π + arcsin ( 0.84697 \dots ) + 2 π 1 )} + 0.7 sin (π + arcsin (0.84697 \dots) + 2 π 1) = 1.1

改良

- 2.28576 \dots = 1.1

\Rightarrow 偽

x = arcsin (0.24522 \dots) + 2 πn, x = arcsin (- 0.84697 \dots) + 2 πn

10進法形式で解を証明する

x = 0.24774 \dots + 2 πn, x = - 1.01026 \dots + 2 πn

人気の例

-sin^2(θ)+cos^2(θ)=1+sin(θ)

- sin^{2} (θ) + cos^{2} (θ) = 1 + sin (θ)

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

cos(θ)sin(θ)=sin(2θ)

cos (θ) sin (θ) = sin (2 θ)

4sin(x)cos(x)=-3cos(x)

4 sin (x) cos (x) = - 3 cos (x)

3sin^2(x)-cos^2(x)=0

3 sin^{2} (x) - cos^{2} (x) = 0