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人気のある三角関数 >

2cot^2(x)-11csc(x)=-14

解

2 cot^{2} (x) - 11 csc (x) = - 14

解

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

2 cot^{2} (x) - 11 csc (x) = - 14

両辺から

- 14

を引く

2 cot^{2} (x) - 11 csc (x) + 14 = 0

三角関数の公式を使用して書き換える

14 - 11 csc (x) + 2 cot^{2} (x)

ピタゴラスの公式を使用する:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= 14 - 11 csc (x) + 2 (csc^{2} (x) - 1)

簡素化

14 - 11 csc (x) + 2 (csc^{2} (x) - 1) : 2 csc^{2} (x) - 11 csc (x) + 12

14 - 11 csc (x) + 2 (csc^{2} (x) - 1)

拡張

2 (csc^{2} (x) - 1) : 2 csc^{2} (x) - 2

2 (csc^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

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

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

数を乗じる:

2 \cdot 1 = 2

= 2 csc^{2} (x) - 2

= 14 - 11 csc (x) + 2 csc^{2} (x) - 2

簡素化

14 - 11 csc (x) + 2 csc^{2} (x) - 2 : 2 csc^{2} (x) - 11 csc (x) + 12

14 - 11 csc (x) + 2 csc^{2} (x) - 2

条件のようなグループ

= - 11 csc (x) + 2 csc^{2} (x) + 14 - 2

数を足す/引く:

14 - 2 = 12

= 2 csc^{2} (x) - 11 csc (x) + 12

= 2 csc^{2} (x) - 11 csc (x) + 12

= 2 csc^{2} (x) - 11 csc (x) + 12

12 - 11 csc (x) + 2 csc^{2} (x) = 0

置換で解く

12 - 11 csc (x) + 2 csc^{2} (x) = 0

仮定:

csc (x) = u

12 - 11 u + 2 u^{2} = 0

12 - 11 u + 2 u^{2} = 0 : u = 4, u = \frac{3}{2}

12 - 11 u + 2 u^{2} = 0

標準的な形式で書く

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

2 u^{2} - 11 u + 12 = 0

解くとthe二次式

2 u^{2} - 11 u + 12 = 0

二次Equationの公式:

次の場合:

a = 2, b = - 11, c = 12

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 2 \cdot 12}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 2 \cdot 12}{2 \cdot 2}

(- 11)^{2} - 4 \cdot 2 \cdot 12 = 5

(- 11)^{2} - 4 \cdot 2 \cdot 12

指数の規則を適用する:

n

が偶数であれば

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

(- 11)^{2} = 1 1^{2}

= 1 1^{2} - 4 \cdot 2 \cdot 12

数を乗じる:

4 \cdot 2 \cdot 12 = 96

= 1 1^{2} - 96

1 1^{2} = 121

= 121 - 96

数を引く:

121 - 96 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 11 ) \pm 5}{2 \cdot 2}

解を分離する

u_{1} = \frac{- ( - 11 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 11 ) - 5}{2 \cdot 2}

u = \frac{- ( - 11 ) + 5}{2 \cdot 2} : 4

\frac{- ( - 11 ) + 5}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{11 + 5}{2 \cdot 2}

数を足す:

11 + 5 = 16

= \frac{16}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{16}{4}

数を割る:

\frac{16}{4} = 4

= 4

u = \frac{- ( - 11 ) - 5}{2 \cdot 2} : \frac{3}{2}

\frac{- ( - 11 ) - 5}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{11 - 5}{2 \cdot 2}

数を引く:

11 - 5 = 6

= \frac{6}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{6}{4}

共通因数を約分する:

2

= \frac{3}{2}

二次equationの解:

u = 4, u = \frac{3}{2}

代用を戻す

u = csc (x)

csc (x) = 4, csc (x) = \frac{3}{2}

csc (x) = 4, csc (x) = \frac{3}{2}

csc (x) = 4 : x = arccsc (4) + 2 πn, x = π - arccsc (4) + 2 πn

csc (x) = 4

三角関数の逆数プロパティを適用する

csc (x) = 4

以下の一般解

csc (x) = 4

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (4) + 2 πn, x = π - arccsc (4) + 2 πn

x = arccsc (4) + 2 πn, x = π - arccsc (4) + 2 πn

csc (x) = \frac{3}{2} : x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

csc (x) = \frac{3}{2}

三角関数の逆数プロパティを適用する

csc (x) = \frac{3}{2}

以下の一般解

csc (x) = \frac{3}{2}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

すべての解を組み合わせる

x = arccsc (4) + 2 πn, x = π - arccsc (4) + 2 πn, x = arccsc (\frac{3}{2}) + 2 πn, x = π - arccsc (\frac{3}{2}) + 2 πn

10進法形式で解を証明する

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

人気の例

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