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人気のある三角関数 >

3sec^2(θ)-2tan^2(θ)-4=0

解

3 sec^{2} (θ) - 2 tan^{2} (θ) - 4 = 0

解

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

+1

度

θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n

解答ステップ

3 sec^{2} (θ) - 2 tan^{2} (θ) - 4 = 0

三角関数の公式を使用して書き換える

- 4 - 2 tan^{2} (θ) + 3 sec^{2} (θ)

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

= - 4 - 2 (sec^{2} (θ) - 1) + 3 sec^{2} (θ)

簡素化

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

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

拡張

- 2 (sec^{2} (θ) - 1) : - 2 sec^{2} (θ) + 2

- 2 (sec^{2} (θ) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = - 2, b = sec^{2} (θ), c = 1

= - 2 sec^{2} (θ) - (- 2) \cdot 1

マイナス・プラスの規則を適用する

- (- a) = a

= - 2 sec^{2} (θ) + 2 \cdot 1

数を乗じる:

2 \cdot 1 = 2

= - 2 sec^{2} (θ) + 2

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

簡素化

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

- 4 - 2 sec^{2} (θ) + 2 + 3 sec^{2} (θ)

条件のようなグループ

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

類似した元を足す:

- 2 sec^{2} (θ) + 3 sec^{2} (θ) = sec^{2} (θ)

= sec^{2} (θ) - 4 + 2

数を足す/引く:

- 4 + 2 = - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

- 2 + sec^{2} (θ) = 0

置換で解く

- 2 + sec^{2} (θ) = 0

仮定:

sec (θ) = u

- 2 + u^{2} = 0

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

- 2 + u^{2} = 0

2

を右側に移動します

- 2 + u^{2} = 0

両辺に

2

を足す

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

簡素化

u^{2} = 2

u^{2} = 2

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = 2, u = - 2

代用を戻す

u = sec (θ)

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2 : θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sec (θ) = 2

以下の一般解

sec (θ) = 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}

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sec (θ) = - 2 : θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

sec (θ) = - 2

以下の一般解

sec (θ) = - 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}

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

すべての解を組み合わせる

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

グラフ

人気の例

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