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人気のある三角関数 >

2sec(t)=tan^2(t)+1

解

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

解

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

+1

度

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

解答ステップ

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

両辺から

tan^{2} (t) + 1

を引く

2 sec (t) - tan^{2} (t) - 1 = 0

三角関数の公式を使用して書き換える

- 1 - tan^{2} (t) + 2 sec (t)

ピタゴラスの公式を使用する:

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

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

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

- sec^{2} (t) + 2 sec (t) = 0

置換で解く

- sec^{2} (t) + 2 sec (t) = 0

仮定:

sec (t) = u

- u^{2} + 2 u = 0

- u^{2} + 2 u = 0 : u = 0, u = 2

- u^{2} + 2 u = 0

解くとthe二次式

- u^{2} + 2 u = 0

二次Equationの公式:

次の場合:

a = - 1, b = 2, c = 0

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 1 ) \cdot 0}{2 ( - 1 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 1 ) \cdot 0}{2 ( - 1 )}

2^{2} - 4 (- 1) \cdot 0 = 2

2^{2} - 4 (- 1) \cdot 0

規則を適用

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 2^{2} + 0

2^{2} + 0 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

, 以下を想定

a \geq 0

= 2

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

解を分離する

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

u = \frac{- 2 + 2}{2 ( - 1 )} : 0

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

括弧を削除する:

(- a) = - a

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

数を足す/引く:

- 2 + 2 = 0

= \frac{0}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{- 2}

分数の規則を適用する:

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

= - \frac{0}{2}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

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

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

括弧を削除する:

(- a) = - a

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

数を引く:

- 2 - 2 = - 4

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

数を乗じる:

2 \cdot 1 = 2

= \frac{- 4}{- 2}

分数の規則を適用する:

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

= \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= 2

二次equationの解:

u = 0, u = 2

代用を戻す

u = sec (t)

sec (t) = 0, sec (t) = 2

sec (t) = 0, sec (t) = 2

sec (t) = 0 :

解なし

sec (t) = 0

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

解なし

sec (t) = 2 : t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

sec (t) = 2

以下の一般解

sec (t) = 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}

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

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

すべての解を組み合わせる

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

グラフ

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