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人気のある三角関数 >

2tan^2(x)+4sec(x)+sec^2(x)=2

解

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

両辺から

2

を引く

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) - 2 = 0

三角関数の公式を使用して書き換える

- 2 + sec^{2} (x) + 2 tan^{2} (x) + 4 sec (x)

ピタゴラスの公式を使用する:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 2 + sec^{2} (x) + 2 (sec^{2} (x) - 1) + 4 sec (x)

簡素化

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

- 2 + sec^{2} (x) + 2 (sec^{2} (x) - 1) + 4 sec (x)

拡張

2 (sec^{2} (x) - 1) : 2 sec^{2} (x) - 2

2 (sec^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

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

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

数を乗じる:

2 \cdot 1 = 2

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

= - 2 + sec^{2} (x) + 2 sec^{2} (x) - 2 + 4 sec (x)

簡素化

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

- 2 + sec^{2} (x) + 2 sec^{2} (x) - 2 + 4 sec (x)

類似した元を足す:

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

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

条件のようなグループ

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

数を引く:

- 2 - 2 = - 4

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

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

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

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

置換で解く

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

仮定:

sec (x) = u

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

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

4^{2} - 4 \cdot 3 (- 4) = 8

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

規則を適用

- (- a) = a

= 4^{2} + 4 \cdot 3 \cdot 4

数を乗じる:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数を足す:

16 + 48 = 64

= 64

数を因数に分解する:

64 = 8^{2}

= 8^{2}

累乗根の規則を適用する:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 3}

解を分離する

u_{1} = \frac{- 4 + 8}{2 \cdot 3}, u_{2} = \frac{- 4 - 8}{2 \cdot 3}

u = \frac{- 4 + 8}{2 \cdot 3} : \frac{2}{3}

\frac{- 4 + 8}{2 \cdot 3}

数を足す/引く:

- 4 + 8 = 4

= \frac{4}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{4}{6}

共通因数を約分する:

2

= \frac{2}{3}

u = \frac{- 4 - 8}{2 \cdot 3} : - 2

\frac{- 4 - 8}{2 \cdot 3}

数を引く:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 12}{6}

分数の規則を適用する:

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

= - \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= - 2

二次equationの解:

u = \frac{2}{3}, u = - 2

代用を戻す

u = sec (x)

sec (x) = \frac{2}{3}, sec (x) = - 2

sec (x) = \frac{2}{3}, sec (x) = - 2

sec (x) = \frac{2}{3} :

解なし

sec (x) = \frac{2}{3}

sec (x) \leq - 1 or sec (x) \geq 1

解なし

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

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

すべての解を組み合わせる

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

グラフ

人気の例

sec(x+10)=5.759

cos(3x+pi/4)=(sqrt(3))/2

tan(θ)=(3sqrt(3))/3

cos(2x)=sqrt(1-sin^2(2x)),0<= x<= 2pi

sqrt(2)=sec(x)tan(x)