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人気のある三角関数 >

5sec^2(x)-5sec(x)=10

解

5 sec^{2} (x) - 5 sec (x) = 10

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

5 sec^{2} (x) - 5 sec (x) = 10

置換で解く

5 sec^{2} (x) - 5 sec (x) = 10

仮定:

sec (x) = u

5 u^{2} - 5 u = 10

5 u^{2} - 5 u = 10 : u = 2, u = - 1

5 u^{2} - 5 u = 10

10

を左側に移動します

5 u^{2} - 5 u = 10

両辺から

10

を引く

5 u^{2} - 5 u - 10 = 10 - 10

簡素化

5 u^{2} - 5 u - 10 = 0

5 u^{2} - 5 u - 10 = 0

解くとthe二次式

5 u^{2} - 5 u - 10 = 0

二次Equationの公式:

次の場合:

a = 5, b = - 5, c = - 10

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 5 ( - 10 )}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 5 ( - 10 )}{2 \cdot 5}

(- 5)^{2} - 4 \cdot 5 (- 10) = 15

(- 5)^{2} - 4 \cdot 5 (- 10)

規則を適用

- (- a) = a

= (- 5)^{2} + 4 \cdot 5 \cdot 10

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 5 \cdot 10

数を乗じる:

4 \cdot 5 \cdot 10 = 200

= 5^{2} + 200

5^{2} = 25

= 25 + 200

数を足す:

25 + 200 = 225

= 225

数を因数に分解する:

225 = 1 5^{2}

= 1 5^{2}

累乗根の規則を適用する:

n a^{n} = a

1 5^{2} = 15

= 15

u_{1, 2} = \frac{- ( - 5 ) \pm 15}{2 \cdot 5}

解を分離する

u_{1} = \frac{- ( - 5 ) + 15}{2 \cdot 5}, u_{2} = \frac{- ( - 5 ) - 15}{2 \cdot 5}

u = \frac{- ( - 5 ) + 15}{2 \cdot 5} : 2

\frac{- ( - 5 ) + 15}{2 \cdot 5}

規則を適用

- (- a) = a

= \frac{5 + 15}{2 \cdot 5}

数を足す:

5 + 15 = 20

= \frac{20}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{20}{10}

数を割る:

\frac{20}{10} = 2

= 2

u = \frac{- ( - 5 ) - 15}{2 \cdot 5} : - 1

\frac{- ( - 5 ) - 15}{2 \cdot 5}

規則を適用

- (- a) = a

= \frac{5 - 15}{2 \cdot 5}

数を引く:

5 - 15 = - 10

= \frac{- 10}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 10}{10}

分数の規則を適用する:

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

= - \frac{10}{10}

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

人気の例

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