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人気のある三角関数 >

sec^2(x)+2=3sec(x)

解

sec^{2} (x) + 2 = 3 sec (x)

解

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

+1

度

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

解答ステップ

sec^{2} (x) + 2 = 3 sec (x)

置換で解く

sec^{2} (x) + 2 = 3 sec (x)

仮定:

sec (x) = u

u^{2} + 2 = 3 u

u^{2} + 2 = 3 u : u = 2, u = 1

u^{2} + 2 = 3 u

3 u

を左側に移動します

u^{2} + 2 = 3 u

両辺から

3 u

を引く

u^{2} + 2 - 3 u = 3 u - 3 u

簡素化

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 1, b = - 3, c = 2

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

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

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

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

指数の規則を適用する:

n

が偶数であれば

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

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

= 3^{2} - 4 \cdot 1 \cdot 2

数を乗じる:

4 \cdot 1 \cdot 2 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数を引く:

9 - 8 = 1

= 1

規則を適用

1 = 1

= 1

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

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を足す:

3 + 1 = 4

= \frac{4}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= 2

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

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

規則を適用

- (- a) = a

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

数を引く:

3 - 1 = 2

= \frac{2}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{2}{2}

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 = 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 = 0 + 2 πn

x = 0 + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 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 = 2 πn

グラフ

人気の例

cos(x)-1=-cos(x)

cos (x) - 1 = - cos (x)

2 cos^{2} (x) = 3

9tan(2x)-9cot(x)=0

9 tan (2 x) - 9 cot (x) = 0

sin(2x)=2sin^2(x)

sin (2 x) = 2 sin^{2} (x)

sin(x)sec(x)=sin(x)

sin (x) sec (x) = sin (x)