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人気のある三角関数 >

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

解

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

解

x = 0.70282 \dots + πn, x = - 0.86797 \dots + πn

+1

度

x = 40.26883 \dots^{\circ} + 18 0^{\circ} n, x = - 49.73116 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

簡素化

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

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

拡張

3 (tan^{2} (x) + 1) : 3 tan^{2} (x) + 3

3 (tan^{2} (x) + 1)

分配法則を適用する:

a (b + c) = ab + a c

a = 3, b = tan^{2} (x), c = 1

= 3 tan^{2} (x) + 3 \cdot 1

数を乗じる:

3 \cdot 1 = 3

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

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

簡素化

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

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

条件のようなグループ

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

数を足す/引く:

- 6 + 3 = - 3

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

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

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

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

置換で解く

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

仮定:

tan (x) = u

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

- 3 + u + 3 u^{2} = 0 : u = \frac{- 1 + 37}{6}, u = \frac{- 1 - 37}{6}

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

1^{2} - 4 \cdot 3 (- 3) = 37

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 3 (- 3)

規則を適用

- (- a) = a

= 1 + 4 \cdot 3 \cdot 3

数を乗じる:

4 \cdot 3 \cdot 3 = 36

= 1 + 36

数を足す:

1 + 36 = 37

= 37

u_{1, 2} = \frac{- 1 \pm 37}{2 \cdot 3}

解を分離する

u_{1} = \frac{- 1 + 37}{2 \cdot 3}, u_{2} = \frac{- 1 - 37}{2 \cdot 3}

u = \frac{- 1 + 37}{2 \cdot 3} : \frac{- 1 + 37}{6}

\frac{- 1 + 37}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 1 + 37}{6}

u = \frac{- 1 - 37}{2 \cdot 3} : \frac{- 1 - 37}{6}

\frac{- 1 - 37}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 1 - 37}{6}

二次equationの解:

u = \frac{- 1 + 37}{6}, u = \frac{- 1 - 37}{6}

代用を戻す

u = tan (x)

tan (x) = \frac{- 1 + 37}{6}, tan (x) = \frac{- 1 - 37}{6}

tan (x) = \frac{- 1 + 37}{6}, tan (x) = \frac{- 1 - 37}{6}

tan (x) = \frac{- 1 + 37}{6} : x = arctan (\frac{- 1 + 37}{6}) + πn

tan (x) = \frac{- 1 + 37}{6}

三角関数の逆数プロパティを適用する

tan (x) = \frac{- 1 + 37}{6}

以下の一般解

tan (x) = \frac{- 1 + 37}{6}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{- 1 + 37}{6}) + πn

x = arctan (\frac{- 1 + 37}{6}) + πn

tan (x) = \frac{- 1 - 37}{6} : x = arctan (\frac{- 1 - 37}{6}) + πn

tan (x) = \frac{- 1 - 37}{6}

三角関数の逆数プロパティを適用する

tan (x) = \frac{- 1 - 37}{6}

以下の一般解

tan (x) = \frac{- 1 - 37}{6}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (\frac{- 1 - 37}{6}) + πn

x = arctan (\frac{- 1 - 37}{6}) + πn

すべての解を組み合わせる

x = arctan (\frac{- 1 + 37}{6}) + πn, x = arctan (\frac{- 1 - 37}{6}) + πn

10進法形式で解を証明する

x = 0.70282 \dots + πn, x = - 0.86797 \dots + πn

グラフ

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