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人気のある三角関数 >

sin(θ)-(cos(θ))/5 =0.6377

解

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

解

θ = 2.66345 \dots + 2 πn, θ = 0.87293 \dots + 2 πn

+1

度

θ = 152.60452 \dots^{\circ} + 36 0^{\circ} n, θ = 50.01533 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

両辺に

\frac{cos ( θ )}{5}

を足す

sin (θ) = 0.6377 + 0.2 cos (θ)

両辺を2乗する

sin^{2} (θ) = (0.6377 + 0.2 cos (θ))^{2}

両辺から

(0.6377 + 0.2 cos (θ))^{2}

を引く

sin^{2} (θ) - 0.40666129 - 0.25508 cos (θ) - 0.04 cos^{2} (θ) = 0

三角関数の公式を使用して書き換える

- 0.40666129 + sin^{2} (θ) - 0.04 cos^{2} (θ) - 0.25508 cos (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)

= - 0.40666129 + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.25508 cos (θ)

簡素化

- 0.40666129 + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.25508 cos (θ) : - 1.04 cos^{2} (θ) - 0.25508 cos (θ) + 0.59333871

- 0.40666129 + 1 - cos^{2} (θ) - 0.04 cos^{2} (θ) - 0.25508 cos (θ)

類似した元を足す:

- cos^{2} (θ) - 0.04 cos^{2} (θ) = - 1.04 cos^{2} (θ)

= - 0.40666129 + 1 - 1.04 cos^{2} (θ) - 0.25508 cos (θ)

数を足す/引く:

- 0.40666129 + 1 = 0.59333871

= - 1.04 cos^{2} (θ) - 0.25508 cos (θ) + 0.59333871

= - 1.04 cos^{2} (θ) - 0.25508 cos (θ) + 0.59333871

0.59333871 - 0.25508 cos (θ) - 1.04 cos^{2} (θ) = 0

置換で解く

0.59333871 - 0.25508 cos (θ) - 1.04 cos^{2} (θ) = 0

仮定:

cos (θ) = u

0.59333871 - 0.25508 u - 1.04 u^{2} = 0

0.59333871 - 0.25508 u - 1.04 u^{2} = 0 : u = - \frac{0.25508 + 2.53335484}{2.08}, u = \frac{2.53335484 - 0.25508}{2.08}

0.59333871 - 0.25508 u - 1.04 u^{2} = 0

標準的な形式で書く

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

- 1.04 u^{2} - 0.25508 u + 0.59333871 = 0

解くとthe二次式

- 1.04 u^{2} - 0.25508 u + 0.59333871 = 0

二次Equationの公式:

次の場合:

a = - 1.04, b = - 0.25508, c = 0.59333871

u_{1, 2} = \frac{- ( - 0.25508 ) \pm ( - 0.25508 ) ^{2} - 4 ( - 1.04 ) \cdot 0.59333871}{2 ( - 1.04 )}

u_{1, 2} = \frac{- ( - 0.25508 ) \pm ( - 0.25508 ) ^{2} - 4 ( - 1.04 ) \cdot 0.59333871}{2 ( - 1.04 )}

(- 0.25508)^{2} - 4 (- 1.04) \cdot 0.59333871 = 2.53335484

(- 0.25508)^{2} - 4 (- 1.04) \cdot 0.59333871

規則を適用

- (- a) = a

= (- 0.25508)^{2} + 4 \cdot 1.04 \cdot 0.59333871

指数の規則を適用する:

n

が偶数であれば

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

(- 0.25508)^{2} = 0.2550 8^{2}

= 0.2550 8^{2} + 4 \cdot 0.59333871 \cdot 1.04

数を乗じる:

4 \cdot 1.04 \cdot 0.59333871 = 2.46828 \dots

= 0.2550 8^{2} + 2.46828 \dots

0.2550 8^{2} = 0.0650658064

= 0.0650658064 + 2.46828 \dots

数を足す:

0.0650658064 + 2.46828 \dots = 2.53335484

= 2.53335484

u_{1, 2} = \frac{- ( - 0.25508 ) \pm 2.53335484}{2 ( - 1.04 )}

解を分離する

u_{1} = \frac{- ( - 0.25508 ) + 2.53335484}{2 ( - 1.04 )}, u_{2} = \frac{- ( - 0.25508 ) - 2.53335484}{2 ( - 1.04 )}

u = \frac{- ( - 0.25508 ) + 2.53335484}{2 ( - 1.04 )} : - \frac{0.25508 + 2.53335484}{2.08}

\frac{- ( - 0.25508 ) + 2.53335484}{2 ( - 1.04 )}

括弧を削除する:

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

= \frac{0.25508 + 2.53335484}{- 2 \cdot 1.04}

数を乗じる:

2 \cdot 1.04 = 2.08

= \frac{0.25508 + 2.53335484}{- 2.08}

分数の規則を適用する:

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

= - \frac{0.25508 + 2.53335484}{2.08}

u = \frac{- ( - 0.25508 ) - 2.53335484}{2 ( - 1.04 )} : \frac{2.53335484 - 0.25508}{2.08}

\frac{- ( - 0.25508 ) - 2.53335484}{2 ( - 1.04 )}

括弧を削除する:

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

= \frac{0.25508 - 2.53335484}{- 2 \cdot 1.04}

数を乗じる:

2 \cdot 1.04 = 2.08

= \frac{0.25508 - 2.53335484}{- 2.08}

分数の規則を適用する:

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

0.25508 - 2.53335484 = - (2.53335484 - 0.25508)

= \frac{2.53335484 - 0.25508}{2.08}

二次equationの解:

u = - \frac{0.25508 + 2.53335484}{2.08}, u = \frac{2.53335484 - 0.25508}{2.08}

代用を戻す

u = cos (θ)

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}, cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}, cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08} : θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}

三角関数の逆数プロパティを適用する

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}

以下の一般解

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}

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

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

cos (θ) = \frac{2.53335484 - 0.25508}{2.08} : θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

三角関数の逆数プロパティを適用する

cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

以下の一般解

cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

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

θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

すべての解を組み合わせる

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

元のequationに当てはめて解を検算する

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

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{0.25508 + 2.53335484}{2.08}) + 2 πn :

真

arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

挿入

n = 1

arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

の挿入向け

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1

sin (arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1) - \frac{cos ( arccos ( - \frac{0.25508 + 2.53335484}{2.08} ) + 2 π 1 )}{5} = 0.6377

改良

0.6377 = 0.6377

\Rightarrow 真

解答を確認する

- arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn :

偽

- arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

挿入

n = 1

- arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

の挿入向け

θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1

sin (- arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 π 1) - \frac{cos ( - arccos ( - \frac{0.25508 + 2.53335484}{2.08} ) + 2 π 1 )}{5} = 0.6377

改良

- 0.28255 \dots = 0.6377

\Rightarrow 偽

解答を確認する

arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn :

真

arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

挿入

n = 1

arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

の挿入向け

θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1

sin (arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1) - \frac{cos ( arccos ( \frac{2.53335484 - 0.25508}{2.08} ) + 2 π 1 )}{5} = 0.6377

改良

0.6377 = 0.6377

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn :

偽

2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

の挿入向け

θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1

sin (2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 π 1) - \frac{cos ( 2 π - arccos ( \frac{2.53335484 - 0.25508}{2.08} ) + 2 π 1 )}{5} = 0.6377

改良

- 0.89473 \dots = 0.6377

\Rightarrow 偽

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

10進法形式で解を証明する

θ = 2.66345 \dots + 2 πn, θ = 0.87293 \dots + 2 πn

グラフ

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