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人気のある三角関数 >

6sec^2(x)-6sec(x)=12

解

6 sec^{2} (x) - 6 sec (x) = 12

解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

6 sec^{2} (x) - 6 sec (x) = 12

置換で解く

6 sec^{2} (x) - 6 sec (x) = 12

仮定:

sec (x) = u

6 u^{2} - 6 u = 12

6 u^{2} - 6 u = 12 : u = 2, u = - 1

6 u^{2} - 6 u = 12

12

を左側に移動します

6 u^{2} - 6 u = 12

両辺から

12

を引く

6 u^{2} - 6 u - 12 = 12 - 12

簡素化

6 u^{2} - 6 u - 12 = 0

6 u^{2} - 6 u - 12 = 0

解くとthe二次式

6 u^{2} - 6 u - 12 = 0

二次Equationの公式:

次の場合:

a = 6, b = - 6, c = - 12

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 6 ( - 12 )}{2 \cdot 6}

(- 6)^{2} - 4 \cdot 6 (- 12) = 18

(- 6)^{2} - 4 \cdot 6 (- 12)

規則を適用

- (- a) = a

= (- 6)^{2} + 4 \cdot 6 \cdot 12

指数の規則を適用する:

n

が偶数であれば

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

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 6 \cdot 12

数を乗じる:

4 \cdot 6 \cdot 12 = 288

= 6^{2} + 288

6^{2} = 36

= 36 + 288

数を足す:

36 + 288 = 324

= 324

数を因数に分解する:

324 = 1 8^{2}

= 1 8^{2}

累乗根の規則を適用する:

n a^{n} = a

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- ( - 6 ) \pm 18}{2 \cdot 6}

解を分離する

u_{1} = \frac{- ( - 6 ) + 18}{2 \cdot 6}, u_{2} = \frac{- ( - 6 ) - 18}{2 \cdot 6}

u = \frac{- ( - 6 ) + 18}{2 \cdot 6} : 2

\frac{- ( - 6 ) + 18}{2 \cdot 6}

規則を適用

- (- a) = a

= \frac{6 + 18}{2 \cdot 6}

数を足す:

6 + 18 = 24

= \frac{24}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{24}{12}

数を割る:

\frac{24}{12} = 2

= 2

u = \frac{- ( - 6 ) - 18}{2 \cdot 6} : - 1

\frac{- ( - 6 ) - 18}{2 \cdot 6}

規則を適用

- (- a) = a

= \frac{6 - 18}{2 \cdot 6}

数を引く:

6 - 18 = - 12

= \frac{- 12}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{- 12}{12}

分数の規則を適用する:

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

= - \frac{12}{12}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

u = 2, u = - 1

代用を戻す

u = sec (x)

sec (x) = 2, sec (x) = - 1

sec (x) = 2, sec (x) = - 1

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = 2

以下の一般解

sec (x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

以下の一般解

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

すべての解を組み合わせる

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

グラフ

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