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人気のある三角関数 >

9.8sin(α)-0.29.8*cos(α)=0.47

解

9.8 \cdot sin (α) - 0.2 \cdot 9.8 \cdot cos (α) = 0.47

解

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

+1

度

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

解答ステップ

9.8 sin (α) - 0.2 \cdot 9.8 cos (α) = 0.47

両辺に

0.29.8 cos (α)

を足す

9.8 sin (α) = 0.47 + 1.96 cos (α)

両辺を2乗する

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

両辺から

(0.47 + 1.96 cos (α))^{2}

を引く

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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))

簡素化

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

拡張

96.04 (1 - cos^{2} (α)) : 96.04 - 96.04 cos^{2} (α)

96.04 (1 - cos^{2} (α))

分配法則を適用する:

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)

数を乗じる:

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)

簡素化

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

条件のようなグループ

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

類似した元を足す:

- 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

数を足す/引く:

- 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

置換で解く

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

仮定:

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

規則を適用

- (- a) = a

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

指数の規則を適用する:

n

が偶数であれば

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

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

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

数を乗じる:

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

数を足す:

3.39443776 + 38282.26007 \dots = 38285.654512

= 38285.654512

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

解を分離する

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 )}

括弧を削除する:

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

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

数を乗じる:

2 \cdot 99.8816 = 199.7632

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

分数の規則を適用する:

\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 )}

括弧を削除する:

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

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

数を乗じる:

2 \cdot 99.8816 = 199.7632

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

分数の規則を適用する:

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

1.8424 - 38285.654512 = - (38285.654512 - 1.8424)

= \frac{38285.654512 - 1.8424}{199.7632}

二次equationの解:

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

代用を戻す

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}

三角関数の逆数プロパティを適用する

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

以下の一般解

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}

三角関数の逆数プロパティを適用する

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

以下の一般解

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

すべての解を組み合わせる

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

元のequationに当てはめて解を検算する

9.8 sin (α) - 0.29.8 cos (α) = 0.47

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

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

偽

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

挿入

n = 1

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

9.8 sin (α) - 0.29.8 cos (α) = 0.47

の挿入向け

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

改良

3.40577 \dots = 0.47

\Rightarrow 偽

解答を確認する

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

真

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

挿入

n = 1

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

9.8 sin (α) - 0.29.8 cos (α) = 0.47

の挿入向け

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

改良

0.47 = 0.47

\Rightarrow 真

解答を確認する

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

真

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

挿入

n = 1

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

9.8 sin (α) - 0.29.8 cos (α) = 0.47

の挿入向け

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

改良

0.47 = 0.47

\Rightarrow 真

解答を確認する

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

偽

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

挿入

n = 1

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

9.8 sin (α) - 0.29.8 cos (α) = 0.47

の挿入向け

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

改良

- 4.27346 \dots = 0.47

\Rightarrow 偽

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

10進法形式で解を証明する

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

グラフ

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