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人気のある三角関数 >

196sin(θ)-49cos(θ)=160

解

196 sin (θ) - 49 cos (θ) = 160

解

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

+1

度

θ = 141.66783 \dots^{\circ} + 36 0^{\circ} n, θ = 66.40464 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

196 sin (θ) - 49 cos (θ) = 160

両辺に

49 cos (θ)

を足す

196 sin (θ) = 160 + 49 cos (θ)

両辺を2乗する

(196 sin (θ))^{2} = (160 + 49 cos (θ))^{2}

両辺から

(160 + 49 cos (θ))^{2}

を引く

38416 sin^{2} (θ) - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) = 0

三角関数の公式を使用して書き換える

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 sin^{2} (θ)

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)

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

簡素化

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ)) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

拡張

38416 (1 - cos^{2} (θ)) : 38416 - 38416 cos^{2} (θ)

38416 (1 - cos^{2} (θ))

分配法則を適用する:

a (b - c) = ab - a c

a = 38416, b = 1, c = cos^{2} (θ)

= 38416 \cdot 1 - 38416 cos^{2} (θ)

数を乗じる:

38416 \cdot 1 = 38416

= 38416 - 38416 cos^{2} (θ)

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

簡素化

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

条件のようなグループ

= - 15680 cos (θ) - 2401 cos^{2} (θ) - 38416 cos^{2} (θ) - 25600 + 38416

類似した元を足す:

- 2401 cos^{2} (θ) - 38416 cos^{2} (θ) = - 40817 cos^{2} (θ)

= - 15680 cos (θ) - 40817 cos^{2} (θ) - 25600 + 38416

数を足す/引く:

- 25600 + 38416 = 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

置換で解く

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

仮定:

cos (θ) = u

12816 - 15680 u - 40817 u^{2} = 0

12816 - 15680 u - 40817 u^{2} = 0 : u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

12816 - 15680 u - 40817 u^{2} = 0

標準的な形式で書く

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

- 40817 u^{2} - 15680 u + 12816 = 0

解くとthe二次式

- 40817 u^{2} - 15680 u + 12816 = 0

二次Equationの公式:

次の場合:

a = - 40817, b = - 15680, c = 12816

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

(- 15680)^{2} - 4 (- 40817) \cdot 12816 = 2338305088

(- 15680)^{2} - 4 (- 40817) \cdot 12816

規則を適用

- (- a) = a

= (- 15680)^{2} + 4 \cdot 40817 \cdot 12816

指数の規則を適用する:

n

が偶数であれば

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

(- 15680)^{2} = 1568 0^{2}

= 1568 0^{2} + 4 \cdot 40817 \cdot 12816

数を乗じる:

4 \cdot 40817 \cdot 12816 = 2092442688

= 1568 0^{2} + 2092442688

1568 0^{2} = 245862400

= 245862400 + 2092442688

数を足す:

245862400 + 2092442688 = 2338305088

= 2338305088

u_{1, 2} = \frac{- ( - 15680 ) \pm 2338305088}{2 ( - 40817 )}

解を分離する

u_{1} = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}, u_{2} = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

u = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )} : - \frac{15680 + 2338305088}{81634}

\frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}

括弧を削除する:

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

= \frac{15680 + 2338305088}{- 2 \cdot 40817}

数を乗じる:

2 \cdot 40817 = 81634

= \frac{15680 + 2338305088}{- 81634}

分数の規則を適用する:

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

= - \frac{15680 + 2338305088}{81634}

u = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )} : \frac{2338305088 - 15680}{81634}

\frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

括弧を削除する:

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

= \frac{15680 - 2338305088}{- 2 \cdot 40817}

数を乗じる:

2 \cdot 40817 = 81634

= \frac{15680 - 2338305088}{- 81634}

分数の規則を適用する:

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

15680 - 2338305088 = - (2338305088 - 15680)

= \frac{2338305088 - 15680}{81634}

二次equationの解:

u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

代用を戻す

u = cos (θ)

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634} : θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = - \frac{15680 + 2338305088}{81634}

三角関数の逆数プロパティを適用する

cos (θ) = - \frac{15680 + 2338305088}{81634}

以下の一般解

cos (θ) = - \frac{15680 + 2338305088}{81634}

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

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634} : θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634}

三角関数の逆数プロパティを適用する

cos (θ) = \frac{2338305088 - 15680}{81634}

以下の一般解

cos (θ) = \frac{2338305088 - 15680}{81634}

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

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

すべての解を組み合わせる

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

元のequationに当てはめて解を検算する

196 sin (θ) - 49 cos (θ) = 160

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{15680 + 2338305088}{81634}) + 2 πn :

真

arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

挿入

n = 1

arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (θ) - 49 cos (θ) = 160

の挿入向け

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

改良

160 = 160

\Rightarrow 真

解答を確認する

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn :

偽

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

挿入

n = 1

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (θ) - 49 cos (θ) = 160

の挿入向け

θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

改良

- 83.12602 \dots = 160

\Rightarrow 偽

解答を確認する

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

真

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

挿入

n = 1

arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (θ) - 49 cos (θ) = 160

の挿入向け

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

改良

160 = 160

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

偽

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (θ) - 49 cos (θ) = 160

の挿入向け

θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

改良

- 199.22691 \dots = 160

\Rightarrow 偽

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

10進法形式で解を証明する

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

グラフ

人気の例

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8 tan (θ) + 15 = 0

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cos (x) = \frac{48.4}{54.5}

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cot(b)=(tan(b)cot(b))/(csc(b))

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