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人気のある三角関数 >

129.8sin(a)-0.6129.8cos(a)=121.79

解

12 \cdot 9.8 \cdot sin (a) - 0.6 \cdot 12 \cdot 9.8 \cdot cos (a) = 12 \cdot 1.79

解

a = - 2.75844 \dots + 2 πn, a = 0.69769 \dots + 2 πn

+1

度

a = - 158.04722 \dots^{\circ} + 36 0^{\circ} n, a = 39.97473 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

12 \cdot 9.8 sin (a) - 0.6 \cdot 12 \cdot 9.8 cos (a) = 12 \cdot 1.79

両辺に

0.6129.8 cos (a)

を足す

117.6 sin (a) = 21.48 + 70.56 cos (a)

両辺を2乗する

(117.6 sin (a))^{2} = (21.48 + 70.56 cos (a))^{2}

両辺から

(21.48 + 70.56 cos (a))^{2}

を引く

13829.76 sin^{2} (a) - 461.3904 - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) = 0

三角関数の公式を使用して書き換える

- 461.3904 + 13829.76 sin^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

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)

= - 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

簡素化

- 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) : - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

- 461.3904 + 13829.76 (1 - cos^{2} (a)) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

拡張

13829.76 (1 - cos^{2} (a)) : 13829.76 - 13829.76 cos^{2} (a)

13829.76 (1 - cos^{2} (a))

分配法則を適用する:

a (b - c) = ab - a c

a = 13829.76, b = 1, c = cos^{2} (a)

= 13829.76 \cdot 1 - 13829.76 cos^{2} (a)

= 1 \cdot 13829.76 - 13829.76 cos^{2} (a)

数を乗じる:

1 \cdot 13829.76 = 13829.76

= 13829.76 - 13829.76 cos^{2} (a)

= - 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

簡素化

- 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) : - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

- 461.3904 + 13829.76 - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a)

条件のようなグループ

= - 13829.76 cos^{2} (a) - 3031.2576 cos (a) - 4978.7136 cos^{2} (a) - 461.3904 + 13829.76

類似した元を足す:

- 13829.76 cos^{2} (a) - 4978.7136 cos^{2} (a) = - 18808.4736 cos^{2} (a)

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) - 461.3904 + 13829.76

数を足す/引く:

- 461.3904 + 13829.76 = 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

= - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) + 13368.3696

13368.3696 - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) = 0

置換で解く

13368.3696 - 18808.4736 cos^{2} (a) - 3031.2576 cos (a) = 0

仮定:

cos (a) = u

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0 : u = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, u = \frac{1014943029.424128 - 3031.2576}{37616.9472}

13368.3696 - 18808.4736 u^{2} - 3031.2576 u = 0

標準的な形式で書く

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

- 18808.4736 u^{2} - 3031.2576 u + 13368.3696 = 0

解くとthe二次式

- 18808.4736 u^{2} - 3031.2576 u + 13368.3696 = 0

二次Equationの公式:

次の場合:

a = - 18808.4736, b = - 3031.2576, c = 13368.3696

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm ( - 3031.2576 ) ^{2} - 4 ( - 18808.4736 ) \cdot 13368.3696}{2 ( - 18808.4736 )}

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm ( - 3031.2576 ) ^{2} - 4 ( - 18808.4736 ) \cdot 13368.3696}{2 ( - 18808.4736 )}

(- 3031.2576)^{2} - 4 (- 18808.4736) \cdot 13368.3696 = 1014943029.424128

(- 3031.2576)^{2} - 4 (- 18808.4736) \cdot 13368.3696

規則を適用

- (- a) = a

= (- 3031.2576)^{2} + 4 \cdot 18808.4736 \cdot 13368.3696

指数の規則を適用する:

n

が偶数であれば

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

(- 3031.2576)^{2} = 3031.257 6^{2}

= 3031.257 6^{2} + 4 \cdot 13368.3696 \cdot 18808.4736

数を乗じる:

4 \cdot 18808.4736 \cdot 13368.3696 = 1005754506.78657 \dots

= 3031.257 6^{2} + 1005754506.78657 \dots

3031.257 6^{2} = 9188522.63755 \dots

= 9188522.63755 \dots + 1005754506.78657 \dots

数を足す:

9188522.63755 \dots + 1005754506.78657 \dots = 1014943029.424128

= 1014943029.424128

u_{1, 2} = \frac{- ( - 3031.2576 ) \pm 1014943029.424128}{2 ( - 18808.4736 )}

解を分離する

u_{1} = \frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )}, u_{2} = \frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )}

u = \frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )} : - \frac{3031.2576 + 1014943029.424128}{37616.9472}

\frac{- ( - 3031.2576 ) + 1014943029.424128}{2 ( - 18808.4736 )}

括弧を削除する:

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

= \frac{3031.2576 + 1014943029.424128}{- 2 \cdot 18808.4736}

数を乗じる:

2 \cdot 18808.4736 = 37616.9472

= \frac{3031.2576 + 1014943029.424128}{- 37616.9472}

分数の規則を適用する:

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

= - \frac{3031.2576 + 1014943029.424128}{37616.9472}

u = \frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )} : \frac{1014943029.424128 - 3031.2576}{37616.9472}

\frac{- ( - 3031.2576 ) - 1014943029.424128}{2 ( - 18808.4736 )}

括弧を削除する:

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

= \frac{3031.2576 - 1014943029.424128}{- 2 \cdot 18808.4736}

数を乗じる:

2 \cdot 18808.4736 = 37616.9472

= \frac{3031.2576 - 1014943029.424128}{- 37616.9472}

分数の規則を適用する:

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

3031.2576 - 1014943029.424128 = - (1014943029.424128 - 3031.2576)

= \frac{1014943029.424128 - 3031.2576}{37616.9472}

二次equationの解:

u = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, u = \frac{1014943029.424128 - 3031.2576}{37616.9472}

代用を戻す

u = cos (a)

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}, cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472} : a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

三角関数の逆数プロパティを適用する

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

以下の一般解

cos (a) = - \frac{3031.2576 + 1014943029.424128}{37616.9472}

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

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472} : a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

三角関数の逆数プロパティを適用する

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

以下の一般解

cos (a) = \frac{1014943029.424128 - 3031.2576}{37616.9472}

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

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

すべての解を組み合わせる

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn, a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

元のequationに当てはめて解を検算する

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

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{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn :

偽

arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

挿入

n = 1

arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

の挿入向け

a = arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

改良

109.40771 \dots = 21.48

\Rightarrow 偽

解答を確認する

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn :

真

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn

挿入

n = 1

- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

の挿入向け

a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (- arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

改良

21.48 = 21.48

\Rightarrow 真

解答を確認する

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn :

真

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

挿入

n = 1

arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

の挿入向け

a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

改良

21.48 = 21.48

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn :

偽

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

129.8 sin (a) - 0.6129.8 cos (a) = 121.79

の挿入向け

a = 2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1

12 \cdot 9.8 sin (2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) - 0.6 \cdot 12 \cdot 9.8 cos (2 π - arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 π 1) = 12 \cdot 1.79

改良

- 129.62418 \dots = 21.48

\Rightarrow 偽

a = - arccos (- \frac{3031.2576 + 1014943029.424128}{37616.9472}) + 2 πn, a = arccos (\frac{1014943029.424128 - 3031.2576}{37616.9472}) + 2 πn

10進法形式で解を証明する

a = - 2.75844 \dots + 2 πn, a = 0.69769 \dots + 2 πn

グラフ

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