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人気のある三角関数 >

21.16=19.6sin(x)-29.4cos(x)

解

21.16 = 19.6 sin (x) - 29.4 cos (x)

解

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

+1

度

x = - 160.47763 \dots^{\circ} + 36 0^{\circ} n, x = 93.09749 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

21.16 = 19.6 sin (x) - 29.4 cos (x)

両辺に

29.4 cos (x)

を足す

19.6 sin (x) = 21.16 + 29.4 cos (x)

両辺を2乗する

(19.6 sin (x))^{2} = (21.16 + 29.4 cos (x))^{2}

両辺から

(21.16 + 29.4 cos (x))^{2}

を引く

384.16 sin^{2} (x) - 447.7456 - 1244.208 cos (x) - 864.36 cos^{2} (x) = 0

三角関数の公式を使用して書き換える

- 447.7456 - 1244.208 cos (x) + 384.16 sin^{2} (x) - 864.36 cos^{2} (x)

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)

= - 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

簡素化

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x) : - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

拡張

384.16 (1 - cos^{2} (x)) : 384.16 - 384.16 cos^{2} (x)

384.16 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 384.16, b = 1, c = cos^{2} (x)

= 384.16 \cdot 1 - 384.16 cos^{2} (x)

= 1 \cdot 384.16 - 384.16 cos^{2} (x)

数を乗じる:

1 \cdot 384.16 = 384.16

= 384.16 - 384.16 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

簡素化

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x) : - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

類似した元を足す:

- 384.16 cos^{2} (x) - 864.36 cos^{2} (x) = - 1248.52 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 1248.52 cos^{2} (x)

条件のようなグループ

= - 1244.208 cos (x) - 1248.52 cos^{2} (x) - 447.7456 + 384.16

数を足す/引く:

- 447.7456 + 384.16 = - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

置換で解く

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

仮定:

cos (x) = u

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0 : u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

標準的な形式で書く

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

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

解くとthe二次式

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

二次Equationの公式:

次の場合:

a = - 1248.52, b = - 1244.208, c = - 63.5856

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856) = 1230501.974016

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856)

規則を適用

- (- a) = a

= (- 1244.208)^{2} - 4 \cdot 1248.52 \cdot 63.5856

指数の規則を適用する:

n

が偶数であれば

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

(- 1244.208)^{2} = 1244.20 8^{2}

= 1244.20 8^{2} - 4 \cdot 63.5856 \cdot 1248.52

数を乗じる:

4 \cdot 1248.52 \cdot 63.5856 = 317551.573248

= 1244.20 8^{2} - 317551.573248

1244.20 8^{2} = 1548053.547264

= 1548053.547264 - 317551.573248

数を引く:

1548053.547264 - 317551.573248 = 1230501.974016

= 1230501.974016

u_{1, 2} = \frac{- ( - 1244.208 ) \pm 1230501.974016}{2 ( - 1248.52 )}

解を分離する

u_{1} = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}, u_{2} = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

u = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 + 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}

括弧を削除する:

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

= \frac{1244.208 + 1230501.974016}{- 2 \cdot 1248.52}

数を乗じる:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 + 1230501.974016}{- 2497.04}

分数の規則を適用する:

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

= - \frac{1244.208 + 1230501.974016}{2497.04}

u = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 - 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

括弧を削除する:

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

= \frac{1244.208 - 1230501.974016}{- 2 \cdot 1248.52}

数を乗じる:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 - 1230501.974016}{- 2497.04}

分数の規則を適用する:

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

= - \frac{1244.208 - 1230501.974016}{2497.04}

二次equationの解:

u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

代用を戻す

u = cos (x)

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

以下の一般解

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

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

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

以下の一般解

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

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

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

すべての解を組み合わせる

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

元のequationに当てはめて解を検算する

19.6 sin (x) - 29.4 cos (x) = 21.16

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{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

偽

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

挿入

n = 1

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

の挿入向け

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

改良

34.25965 \dots = 21.16

\Rightarrow 偽

解答を確認する

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

真

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

挿入

n = 1

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

の挿入向け

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

改良

21.16 = 21.16

\Rightarrow 真

解答を確認する

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

真

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

挿入

n = 1

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

の挿入向け

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

改良

21.16 = 21.16

\Rightarrow 真

解答を確認する

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

偽

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

挿入

n = 1

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

の挿入向け

x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

改良

- 17.98273 \dots = 21.16

\Rightarrow 偽

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

10進法形式で解を証明する

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

グラフ

人気の例

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

(2 cos (x)) - 1 = 0

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

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

sin(θ)=(5pi)/3

sin (θ) = \frac{5 π}{3}

10 cos (θ) = 8