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人気のある三角関数 >

4tan^2(x)+sec(x)=-2tan^2(x)+sec^2(x)-3

解

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

解

以下の解はない : x \in R

解答ステップ

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

両辺から

- 2 tan^{2} (x) + sec^{2} (x) - 3

を引く

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

三角関数の公式を使用して書き換える

3 + sec (x) - sec^{2} (x) + 6 tan^{2} (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

= 3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

簡素化

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1) : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

拡張

6 (sec^{2} (x) - 1) : 6 sec^{2} (x) - 6

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

分配法則を適用する:

a (b - c) = ab - a c

a = 6, b = sec^{2} (x), c = 1

= 6 sec^{2} (x) - 6 \cdot 1

数を乗じる:

6 \cdot 1 = 6

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

= 3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

簡素化

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6 : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

類似した元を足す:

- sec^{2} (x) + 6 sec^{2} (x) = 5 sec^{2} (x)

= 3 + sec (x) + 5 sec^{2} (x) - 6

条件のようなグループ

= sec (x) + 5 sec^{2} (x) + 3 - 6

数を足す/引く:

3 - 6 = - 3

= 5 sec^{2} (x) + sec (x) - 3

= 5 sec^{2} (x) + sec (x) - 3

= 5 sec^{2} (x) + sec (x) - 3

- 3 + sec (x) + 5 sec^{2} (x) = 0

置換で解く

- 3 + sec (x) + 5 sec^{2} (x) = 0

仮定:

sec (x) = u

- 3 + u + 5 u^{2} = 0

- 3 + u + 5 u^{2} = 0 : u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

- 3 + u + 5 u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 5, b = 1, c = - 3

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

1^{2} - 4 \cdot 5 (- 3) = 61

1^{2} - 4 \cdot 5 (- 3)

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 5 (- 3)

規則を適用

- (- a) = a

= 1 + 4 \cdot 5 \cdot 3

数を乗じる:

4 \cdot 5 \cdot 3 = 60

= 1 + 60

数を足す:

1 + 60 = 61

= 61

u_{1, 2} = \frac{- 1 \pm 61}{2 \cdot 5}

解を分離する

u_{1} = \frac{- 1 + 61}{2 \cdot 5}, u_{2} = \frac{- 1 - 61}{2 \cdot 5}

u = \frac{- 1 + 61}{2 \cdot 5} : \frac{- 1 + 61}{10}

\frac{- 1 + 61}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 1 + 61}{10}

u = \frac{- 1 - 61}{2 \cdot 5} : \frac{- 1 - 61}{10}

\frac{- 1 - 61}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 1 - 61}{10}

二次equationの解:

u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

代用を戻す

u = sec (x)

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10} :

解なし

sec (x) = \frac{- 1 + 61}{10}

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

解なし

sec (x) = \frac{- 1 - 61}{10} :

解なし

sec (x) = \frac{- 1 - 61}{10}

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

解なし

すべての解を組み合わせる

以下の解はない : x \in R

グラフ

人気の例

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