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人気のある三角関数 >

3cot^2(x)=7csc(x)-7

解

3 cot^{2} (x) = 7 csc (x) - 7

解

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

+1

度

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

解答ステップ

3 cot^{2} (x) = 7 csc (x) - 7

両辺から

7 csc (x) - 7

を引く

3 cot^{2} (x) - 7 csc (x) + 7 = 0

三角関数の公式を使用して書き換える

7 + 3 cot^{2} (x) - 7 csc (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

= 7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

簡素化

7 + 3 (csc^{2} (x) - 1) - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

拡張

3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 3

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

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

= 3 csc^{2} (x) - 3 \cdot 1

数を乗じる:

3 \cdot 1 = 3

= 3 csc^{2} (x) - 3

= 7 + 3 csc^{2} (x) - 3 - 7 csc (x)

簡素化

7 + 3 csc^{2} (x) - 3 - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 csc^{2} (x) - 3 - 7 csc (x)

条件のようなグループ

= 3 csc^{2} (x) - 7 csc (x) + 7 - 3

数を足す/引く:

7 - 3 = 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

4 + 3 csc^{2} (x) - 7 csc (x) = 0

置換で解く

4 + 3 csc^{2} (x) - 7 csc (x) = 0

仮定:

csc (x) = u

4 + 3 u^{2} - 7 u = 0

4 + 3 u^{2} - 7 u = 0 : u = \frac{4}{3}, u = 1

4 + 3 u^{2} - 7 u = 0

標準的な形式で書く

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

3 u^{2} - 7 u + 4 = 0

解くとthe二次式

3 u^{2} - 7 u + 4 = 0

二次Equationの公式:

次の場合:

a = 3, b = - 7, c = 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

指数の規則を適用する:

n

が偶数であれば

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

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

数を乗じる:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数を引く:

49 - 48 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

解を分離する

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

u = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

数を足す:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{8}{6}

共通因数を約分する:

2

= \frac{4}{3}

u = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

数を引く:

7 - 1 = 6

= \frac{6}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{6}{6}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

u = \frac{4}{3}, u = 1

代用を戻す

u = csc (x)

csc (x) = \frac{4}{3}, csc (x) = 1

csc (x) = \frac{4}{3}, csc (x) = 1

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

以下の一般解

csc (x) = 1

csc (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

グラフ

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