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人気のある三角関数 >

9sec^2(x)-9sec(x)=18

解

9 sec^{2} (x) - 9 sec (x) = 18

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

9 sec^{2} (x) - 9 sec (x) = 18

置換で解く

9 sec^{2} (x) - 9 sec (x) = 18

仮定:

sec (x) = u

9 u^{2} - 9 u = 18

9 u^{2} - 9 u = 18 : u = 2, u = - 1

9 u^{2} - 9 u = 18

18

を左側に移動します

9 u^{2} - 9 u = 18

両辺から

18

を引く

9 u^{2} - 9 u - 18 = 18 - 18

簡素化

9 u^{2} - 9 u - 18 = 0

9 u^{2} - 9 u - 18 = 0

解くとthe二次式

9 u^{2} - 9 u - 18 = 0

二次Equationの公式:

次の場合:

a = 9, b = - 9, c = - 18

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 9 ( - 18 )}{2 \cdot 9}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 9 ( - 18 )}{2 \cdot 9}

(- 9)^{2} - 4 \cdot 9 (- 18) = 27

(- 9)^{2} - 4 \cdot 9 (- 18)

規則を適用

- (- a) = a

= (- 9)^{2} + 4 \cdot 9 \cdot 18

指数の規則を適用する:

n

が偶数であれば

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

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 9 \cdot 18

数を乗じる:

4 \cdot 9 \cdot 18 = 648

= 9^{2} + 648

9^{2} = 81

= 81 + 648

数を足す:

81 + 648 = 729

= 729

数を因数に分解する:

729 = 2 7^{2}

= 2 7^{2}

累乗根の規則を適用する:

n a^{n} = a

2 7^{2} = 27

= 27

u_{1, 2} = \frac{- ( - 9 ) \pm 27}{2 \cdot 9}

解を分離する

u_{1} = \frac{- ( - 9 ) + 27}{2 \cdot 9}, u_{2} = \frac{- ( - 9 ) - 27}{2 \cdot 9}

u = \frac{- ( - 9 ) + 27}{2 \cdot 9} : 2

\frac{- ( - 9 ) + 27}{2 \cdot 9}

規則を適用

- (- a) = a

= \frac{9 + 27}{2 \cdot 9}

数を足す:

9 + 27 = 36

= \frac{36}{2 \cdot 9}

数を乗じる:

2 \cdot 9 = 18

= \frac{36}{18}

数を割る:

\frac{36}{18} = 2

= 2

u = \frac{- ( - 9 ) - 27}{2 \cdot 9} : - 1

\frac{- ( - 9 ) - 27}{2 \cdot 9}

規則を適用

- (- a) = a

= \frac{9 - 27}{2 \cdot 9}

数を引く:

9 - 27 = - 18

= \frac{- 18}{2 \cdot 9}

数を乗じる:

2 \cdot 9 = 18

= \frac{- 18}{18}

分数の規則を適用する:

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

= - \frac{18}{18}

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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