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人気のある三角関数 >

sin(x)-0.5*cos(x)=0.768

解

sin (x) - 0.5 \cdot cos (x) = 0.768

解

x = 2.84799 \dots + 2 πn, x = 1.22089 \dots + 2 πn

+1

度

x = 163.17825 \dots^{\circ} + 36 0^{\circ} n, x = 69.95184 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin (x) - 0.5 cos (x) = 0.768

両辺に

0.5 cos (x)

を足す

sin (x) = 0.768 + 0.5 cos (x)

両辺を2乗する

sin^{2} (x) = (0.768 + 0.5 cos (x))^{2}

両辺から

(0.768 + 0.5 cos (x))^{2}

を引く

sin^{2} (x) - 0.589824 - 0.768 cos (x) - 0.25 cos^{2} (x) = 0

三角関数の公式を使用して書き換える

- 0.589824 + sin^{2} (x) - 0.25 cos^{2} (x) - 0.768 cos (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)

= - 0.589824 + 1 - cos^{2} (x) - 0.25 cos^{2} (x) - 0.768 cos (x)

簡素化

- 0.589824 + 1 - cos^{2} (x) - 0.25 cos^{2} (x) - 0.768 cos (x) : - 1.25 cos^{2} (x) - 0.768 cos (x) + 0.410176

- 0.589824 + 1 - cos^{2} (x) - 0.25 cos^{2} (x) - 0.768 cos (x)

類似した元を足す:

- cos^{2} (x) - 0.25 cos^{2} (x) = - 1.25 cos^{2} (x)

= - 0.589824 + 1 - 1.25 cos^{2} (x) - 0.768 cos (x)

数を足す/引く:

- 0.589824 + 1 = 0.410176

= - 1.25 cos^{2} (x) - 0.768 cos (x) + 0.410176

= - 1.25 cos^{2} (x) - 0.768 cos (x) + 0.410176

0.410176 - 0.768 cos (x) - 1.25 cos^{2} (x) = 0

置換で解く

0.410176 - 0.768 cos (x) - 1.25 cos^{2} (x) = 0

仮定:

cos (x) = u

0.410176 - 0.768 u - 1.25 u^{2} = 0

0.410176 - 0.768 u - 1.25 u^{2} = 0 : u = - \frac{0.768 + 2.640704}{2.5}, u = \frac{2.640704 - 0.768}{2.5}

0.410176 - 0.768 u - 1.25 u^{2} = 0

標準的な形式で書く

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

- 1.25 u^{2} - 0.768 u + 0.410176 = 0

解くとthe二次式

- 1.25 u^{2} - 0.768 u + 0.410176 = 0

二次Equationの公式:

次の場合:

a = - 1.25, b = - 0.768, c = 0.410176

u_{1, 2} = \frac{- ( - 0.768 ) \pm ( - 0.768 ) ^{2} - 4 ( - 1.25 ) \cdot 0.410176}{2 ( - 1.25 )}

u_{1, 2} = \frac{- ( - 0.768 ) \pm ( - 0.768 ) ^{2} - 4 ( - 1.25 ) \cdot 0.410176}{2 ( - 1.25 )}

(- 0.768)^{2} - 4 (- 1.25) \cdot 0.410176 = 2.640704

(- 0.768)^{2} - 4 (- 1.25) \cdot 0.410176

規則を適用

- (- a) = a

= (- 0.768)^{2} + 4 \cdot 1.25 \cdot 0.410176

指数の規則を適用する:

n

が偶数であれば

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

(- 0.768)^{2} = 0.76 8^{2}

= 0.76 8^{2} + 4 \cdot 0.410176 \cdot 1.25

数を乗じる:

4 \cdot 1.25 \cdot 0.410176 = 2.05088

= 0.76 8^{2} + 2.05088

0.76 8^{2} = 0.589824

= 0.589824 + 2.05088

数を足す:

0.589824 + 2.05088 = 2.640704

= 2.640704

u_{1, 2} = \frac{- ( - 0.768 ) \pm 2.640704}{2 ( - 1.25 )}

解を分離する

u_{1} = \frac{- ( - 0.768 ) + 2.640704}{2 ( - 1.25 )}, u_{2} = \frac{- ( - 0.768 ) - 2.640704}{2 ( - 1.25 )}

u = \frac{- ( - 0.768 ) + 2.640704}{2 ( - 1.25 )} : - \frac{0.768 + 2.640704}{2.5}

\frac{- ( - 0.768 ) + 2.640704}{2 ( - 1.25 )}

括弧を削除する:

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

= \frac{0.768 + 2.640704}{- 2 \cdot 1.25}

数を乗じる:

2 \cdot 1.25 = 2.5

= \frac{0.768 + 2.640704}{- 2.5}

分数の規則を適用する:

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

= - \frac{0.768 + 2.640704}{2.5}

u = \frac{- ( - 0.768 ) - 2.640704}{2 ( - 1.25 )} : \frac{2.640704 - 0.768}{2.5}

\frac{- ( - 0.768 ) - 2.640704}{2 ( - 1.25 )}

括弧を削除する:

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

= \frac{0.768 - 2.640704}{- 2 \cdot 1.25}

数を乗じる:

2 \cdot 1.25 = 2.5

= \frac{0.768 - 2.640704}{- 2.5}

分数の規則を適用する:

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

0.768 - 2.640704 = - (2.640704 - 0.768)

= \frac{2.640704 - 0.768}{2.5}

二次equationの解:

u = - \frac{0.768 + 2.640704}{2.5}, u = \frac{2.640704 - 0.768}{2.5}

代用を戻す

u = cos (x)

cos (x) = - \frac{0.768 + 2.640704}{2.5}, cos (x) = \frac{2.640704 - 0.768}{2.5}

cos (x) = - \frac{0.768 + 2.640704}{2.5}, cos (x) = \frac{2.640704 - 0.768}{2.5}

cos (x) = - \frac{0.768 + 2.640704}{2.5} : x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = - arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn

cos (x) = - \frac{0.768 + 2.640704}{2.5}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{0.768 + 2.640704}{2.5}

以下の一般解

cos (x) = - \frac{0.768 + 2.640704}{2.5}

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

x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = - arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn

x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = - arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn

cos (x) = \frac{2.640704 - 0.768}{2.5} : x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn, x = 2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

cos (x) = \frac{2.640704 - 0.768}{2.5}

三角関数の逆数プロパティを適用する

cos (x) = \frac{2.640704 - 0.768}{2.5}

以下の一般解

cos (x) = \frac{2.640704 - 0.768}{2.5}

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

x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn, x = 2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn, x = 2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

すべての解を組み合わせる

x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = - arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn, x = 2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

元のequationに当てはめて解を検算する

sin (x) - 0.5 cos (x) = 0.768

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{0.768 + 2.640704}{2.5}) + 2 πn :

真

arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn

挿入

n = 1

arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1

sin (x) - 0.5 cos (x) = 0.768

の挿入向け

x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1

sin (arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1) - 0.5 cos (arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1) = 0.768

改良

0.768 = 0.768

\Rightarrow 真

解答を確認する

- arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn :

偽

- arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn

挿入

n = 1

- arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1

sin (x) - 0.5 cos (x) = 0.768

の挿入向け

x = - arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1

sin (- arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1) - 0.5 cos (- arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 π 1) = 0.768

改良

0.18920 \dots = 0.768

\Rightarrow 偽

解答を確認する

arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn :

真

arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

挿入

n = 1

arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1

sin (x) - 0.5 cos (x) = 0.768

の挿入向け

x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1

sin (arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1) - 0.5 cos (arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1) = 0.768

改良

0.768 = 0.768

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn :

偽

2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1

sin (x) - 0.5 cos (x) = 0.768

の挿入向け

x = 2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1

sin (2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1) - 0.5 cos (2 π - arccos (\frac{2.640704 - 0.768}{2.5}) + 2 π 1) = 0.768

改良

- 1.11080 \dots = 0.768

\Rightarrow 偽

x = arccos (- \frac{0.768 + 2.640704}{2.5}) + 2 πn, x = arccos (\frac{2.640704 - 0.768}{2.5}) + 2 πn

10進法形式で解を証明する

x = 2.84799 \dots + 2 πn, x = 1.22089 \dots + 2 πn

グラフ

人気の例

sin(x/2)= 1/2 ,0<= ,x<= 360

sin (\frac{x}{2}) = \frac{1}{2}, 0 \leq, x \leq 36 0^{\circ}

tan (x) = \frac{4}{10}

solvefor x,sin(y/x)+cos(x/y)=0

so l v e f or x, sin (\frac{y}{x}) + cos (\frac{x}{y}) = 0

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

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

solvefor θ,y=4sin(θ)

so l v e f or θ, y = 4 sin (θ)