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人気のある三角関数 >

sec^2(x)+4sec(x)-12=0

解

sec^{2} (x) + 4 sec (x) - 12 = 0

解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 99.59406 \dots^{\circ} + 36 0^{\circ} n, x = - 99.59406 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sec^{2} (x) + 4 sec (x) - 12 = 0

置換で解く

sec^{2} (x) + 4 sec (x) - 12 = 0

仮定:

sec (x) = u

u^{2} + 4 u - 12 = 0

u^{2} + 4 u - 12 = 0 : u = 2, u = - 6

u^{2} + 4 u - 12 = 0

解くとthe二次式

u^{2} + 4 u - 12 = 0

二次Equationの公式:

次の場合:

a = 1, b = 4, c = - 12

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

4^{2} - 4 \cdot 1 \cdot (- 12) = 8

4^{2} - 4 \cdot 1 \cdot (- 12)

規則を適用

- (- a) = a

= 4^{2} + 4 \cdot 1 \cdot 12

数を乗じる:

4 \cdot 1 \cdot 12 = 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:

n a^{n} = a

8^{2} = 8

= 8

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

解を分離する

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

u = \frac{- 4 + 8}{2 \cdot 1} : 2

\frac{- 4 + 8}{2 \cdot 1}

数を足す/引く:

- 4 + 8 = 4

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

数を乗じる:

2 \cdot 1 = 2

= \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= 2

u = \frac{- 4 - 8}{2 \cdot 1} : - 6

\frac{- 4 - 8}{2 \cdot 1}

数を引く:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 12}{2}

分数の規則を適用する:

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

= - \frac{12}{2}

数を割る:

\frac{12}{2} = 6

= - 6

二次equationの解:

u = 2, u = - 6

代用を戻す

u = sec (x)

sec (x) = 2, sec (x) = - 6

sec (x) = 2, sec (x) = - 6

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

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

sec (x) = - 6 : x = arcsec (- 6) + 2 πn, x = - arcsec (- 6) + 2 πn

sec (x) = - 6

三角関数の逆数プロパティを適用する

sec (x) = - 6

以下の一般解

sec (x) = - 6

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- 6) + 2 πn, x = - arcsec (- 6) + 2 πn

x = arcsec (- 6) + 2 πn, x = - arcsec (- 6) + 2 πn

すべての解を組み合わせる

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = arcsec (- 6) + 2 πn, x = - arcsec (- 6) + 2 πn

10進法形式で解を証明する

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

グラフ

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