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人気のある >

0.1cos(x)+0.45sin(x)=0.3125

解

0.1 cos (x) + 0.45 sin (x) = 0.3125

解

x = 0.52624 \dots + 2 πn, x = π - 0.96358 \dots + 2 πn

+1

度

x = 30.15154 \dots^{\circ} + 36 0^{\circ} n, x = 124.79083 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

0.1 cos (x) + 0.45 sin (x) = 0.3125

両辺から

0.45 sin (x)

を引く

0.1 cos (x) = 0.3125 - 0.45 sin (x)

両辺を2乗する

(0.1 cos (x))^{2} = (0.3125 - 0.45 sin (x))^{2}

両辺から

(0.3125 - 0.45 sin (x))^{2}

を引く

0.01 cos^{2} (x) - 0.09765625 + 0.28125 sin (x) - 0.2025 sin^{2} (x) = 0

三角関数の公式を使用して書き換える

- 0.09765625 + 0.01 cos^{2} (x) - 0.2025 sin^{2} (x) + 0.28125 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)

= - 0.09765625 + 0.01 (1 - sin^{2} (x)) - 0.2025 sin^{2} (x) + 0.28125 sin (x)

簡素化

- 0.09765625 + 0.01 (1 - sin^{2} (x)) - 0.2025 sin^{2} (x) + 0.28125 sin (x) : 0.28125 sin (x) - 0.2125 sin^{2} (x) - 0.08765625

- 0.09765625 + 0.01 (1 - sin^{2} (x)) - 0.2025 sin^{2} (x) + 0.28125 sin (x)

拡張

0.01 (1 - sin^{2} (x)) : 0.01 - 0.01 sin^{2} (x)

0.01 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 0.01, b = 1, c = sin^{2} (x)

= 0.01 \cdot 1 - 0.01 sin^{2} (x)

= 1 \cdot 0.01 - 0.01 sin^{2} (x)

数を乗じる:

1 \cdot 0.01 = 0.01

= 0.01 - 0.01 sin^{2} (x)

= - 0.09765625 + 0.01 - 0.01 sin^{2} (x) - 0.2025 sin^{2} (x) + 0.28125 sin (x)

簡素化

- 0.09765625 + 0.01 - 0.01 sin^{2} (x) - 0.2025 sin^{2} (x) + 0.28125 sin (x) : 0.28125 sin (x) - 0.2125 sin^{2} (x) - 0.08765625

- 0.09765625 + 0.01 - 0.01 sin^{2} (x) - 0.2025 sin^{2} (x) + 0.28125 sin (x)

類似した元を足す:

- 0.01 sin^{2} (x) - 0.2025 sin^{2} (x) = - 0.2125 sin^{2} (x)

= - 0.09765625 + 0.01 - 0.2125 sin^{2} (x) + 0.28125 sin (x)

数を足す/引く:

- 0.09765625 + 0.01 = - 0.08765625

= 0.28125 sin (x) - 0.2125 sin^{2} (x) - 0.08765625

= 0.28125 sin (x) - 0.2125 sin^{2} (x) - 0.08765625

= 0.28125 sin (x) - 0.2125 sin^{2} (x) - 0.08765625

- 0.08765625 - 0.2125 sin^{2} (x) + 0.28125 sin (x) = 0

置換で解く

- 0.08765625 - 0.2125 sin^{2} (x) + 0.28125 sin (x) = 0

仮定:

sin (x) = u

- 0.08765625 - 0.2125 u^{2} + 0.28125 u = 0

- 0.08765625 - 0.2125 u^{2} + 0.28125 u = 0 : u = \frac{0.28125 - 0.00459375}{0.425}, u = \frac{0.28125 + 0.00459375}{0.425}

- 0.08765625 - 0.2125 u^{2} + 0.28125 u = 0

標準的な形式で書く

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

- 0.2125 u^{2} + 0.28125 u - 0.08765625 = 0

解くとthe二次式

- 0.2125 u^{2} + 0.28125 u - 0.08765625 = 0

二次Equationの公式:

次の場合:

a = - 0.2125, b = 0.28125, c = - 0.08765625

u_{1, 2} = \frac{- 0.28125 \pm 0.2812 5 ^{2} - 4 ( - 0.2125 ) ( - 0.08765625 )}{2 ( - 0.2125 )}

u_{1, 2} = \frac{- 0.28125 \pm 0.2812 5 ^{2} - 4 ( - 0.2125 ) ( - 0.08765625 )}{2 ( - 0.2125 )}

0.2812 5^{2} - 4 (- 0.2125) (- 0.08765625) = 0.00459375

0.2812 5^{2} - 4 (- 0.2125) (- 0.08765625)

規則を適用

- (- a) = a

= 0.2812 5^{2} - 4 \cdot 0.2125 \cdot 0.08765625

数を乗じる:

4 \cdot 0.2125 \cdot 0.08765625 = 0.0745078125

= 0.2812 5^{2} - 0.0745078125

0.2812 5^{2} = 0.0791015625

= 0.0791015625 - 0.07450 \dots

数を引く:

0.0791015625 - 0.07450 \dots = 0.00459375

= 0.00459375

u_{1, 2} = \frac{- 0.28125 \pm 0.00459375}{2 ( - 0.2125 )}

解を分離する

u_{1} = \frac{- 0.28125 + 0.00459375}{2 ( - 0.2125 )}, u_{2} = \frac{- 0.28125 - 0.00459375}{2 ( - 0.2125 )}

u = \frac{- 0.28125 + 0.00459375}{2 ( - 0.2125 )} : \frac{0.28125 - 0.00459375}{0.425}

\frac{- 0.28125 + 0.00459375}{2 ( - 0.2125 )}

括弧を削除する:

(- a) = - a

= \frac{- 0.28125 + 0.00459375}{- 2 \cdot 0.2125}

数を乗じる:

2 \cdot 0.2125 = 0.425

= \frac{- 0.28125 + 0.00459375}{- 0.425}

分数の規則を適用する:

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

- 0.28125 + 0.00459375 = - (0.28125 - 0.00459375)

= \frac{0.28125 - 0.00459375}{0.425}

u = \frac{- 0.28125 - 0.00459375}{2 ( - 0.2125 )} : \frac{0.28125 + 0.00459375}{0.425}

\frac{- 0.28125 - 0.00459375}{2 ( - 0.2125 )}

括弧を削除する:

(- a) = - a

= \frac{- 0.28125 - 0.00459375}{- 2 \cdot 0.2125}

数を乗じる:

2 \cdot 0.2125 = 0.425

= \frac{- 0.28125 - 0.00459375}{- 0.425}

分数の規則を適用する:

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

- 0.28125 - 0.00459375 = - (0.28125 + 0.00459375)

= \frac{0.28125 + 0.00459375}{0.425}

二次equationの解:

u = \frac{0.28125 - 0.00459375}{0.425}, u = \frac{0.28125 + 0.00459375}{0.425}

代用を戻す

u = sin (x)

sin (x) = \frac{0.28125 - 0.00459375}{0.425}, sin (x) = \frac{0.28125 + 0.00459375}{0.425}

sin (x) = \frac{0.28125 - 0.00459375}{0.425}, sin (x) = \frac{0.28125 + 0.00459375}{0.425}

sin (x) = \frac{0.28125 - 0.00459375}{0.425} : x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn

sin (x) = \frac{0.28125 - 0.00459375}{0.425}

三角関数の逆数プロパティを適用する

sin (x) = \frac{0.28125 - 0.00459375}{0.425}

以下の一般解

sin (x) = \frac{0.28125 - 0.00459375}{0.425}

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

x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn

x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn

sin (x) = \frac{0.28125 + 0.00459375}{0.425} : x = arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

sin (x) = \frac{0.28125 + 0.00459375}{0.425}

三角関数の逆数プロパティを適用する

sin (x) = \frac{0.28125 + 0.00459375}{0.425}

以下の一般解

sin (x) = \frac{0.28125 + 0.00459375}{0.425}

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

x = arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

x = arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

元のequationに当てはめて解を検算する

0.1 cos (x) + 0.45 sin (x) = 0.3125

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 (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn :

真

arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn

挿入

n = 1

arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1

0.1 cos (x) + 0.45 sin (x) = 0.3125

の挿入向け

x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1

0.1 cos (arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1) + 0.45 sin (arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1) = 0.3125

改良

0.3125 = 0.3125

\Rightarrow 真

解答を確認する

π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn :

偽

π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn

挿入

n = 1

π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1

0.1 cos (x) + 0.45 sin (x) = 0.3125

の挿入向け

x = π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1

0.1 cos (π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1) + 0.45 sin (π - arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 π 1) = 0.3125

改良

0.13956 \dots = 0.3125

\Rightarrow 偽

解答を確認する

arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn :

偽

arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

挿入

n = 1

arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1

0.1 cos (x) + 0.45 sin (x) = 0.3125

の挿入向け

x = arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1

0.1 cos (arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1) + 0.45 sin (arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1) = 0.3125

改良

0.42661 \dots = 0.3125

\Rightarrow 偽

解答を確認する

π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn :

真

π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

挿入

n = 1

π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1

0.1 cos (x) + 0.45 sin (x) = 0.3125

の挿入向け

x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1

0.1 cos (π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1) + 0.45 sin (π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 π 1) = 0.3125

改良

0.3125 = 0.3125

\Rightarrow 真

x = arcsin (\frac{0.28125 - 0.00459375}{0.425}) + 2 πn, x = π - arcsin (\frac{0.28125 + 0.00459375}{0.425}) + 2 πn

10進法形式で解を証明する

x = 0.52624 \dots + 2 πn, x = π - 0.96358 \dots + 2 πn

人気の例

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