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人気のある三角関数 >

sec^2(x)-8sec(-x)-20=0

解

sec^{2} (x) - 8 sec (- x) - 20 = 0

解

x = 1.47062 \dots + 2 πn, x = 2 π - 1.47062 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度

x = 84.26082 \dots^{\circ} + 36 0^{\circ} n, x = 275.73917 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

解答ステップ

sec^{2} (x) - 8 sec (- x) - 20 = 0

三角関数の公式を使用して書き換える

sec^{2} (x) - 8 sec (- x) - 20 = 0

- 20 + sec^{2} (x) - 8 sec (x) = 0

- 20 + sec^{2} (x) - 8 sec (x) = 0

置換で解く

- 20 + sec^{2} (x) - 8 sec (x) = 0

仮定:

sec (x) = u

- 20 + u^{2} - 8 u = 0

- 20 + u^{2} - 8 u = 0 : u = 10, u = - 2

- 20 + u^{2} - 8 u = 0

標準的な形式で書く

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

u^{2} - 8 u - 20 = 0

解くとthe二次式

u^{2} - 8 u - 20 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 8, c = - 20

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot ( - 20 )}{2 \cdot 1}

(- 8)^{2} - 4 \cdot 1 \cdot (- 20) = 12

(- 8)^{2} - 4 \cdot 1 \cdot (- 20)

規則を適用

- (- a) = a

= (- 8)^{2} + 4 \cdot 1 \cdot 20

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 1 \cdot 20

数を乗じる:

4 \cdot 1 \cdot 20 = 80

= 8^{2} + 80

8^{2} = 64

= 64 + 80

数を足す:

64 + 80 = 144

= 144

数を因数に分解する:

144 = 1 2^{2}

= 1 2^{2}

累乗根の規則を適用する:

n a^{n} = a

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- ( - 8 ) \pm 12}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 8 ) + 12}{2 \cdot 1}, u_{2} = \frac{- ( - 8 ) - 12}{2 \cdot 1}

u = \frac{- ( - 8 ) + 12}{2 \cdot 1} : 10

\frac{- ( - 8 ) + 12}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{8 + 12}{2 \cdot 1}

数を足す:

8 + 12 = 20

= \frac{20}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{20}{2}

数を割る:

\frac{20}{2} = 10

= 10

u = \frac{- ( - 8 ) - 12}{2 \cdot 1} : - 2

\frac{- ( - 8 ) - 12}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{8 - 12}{2 \cdot 1}

数を引く:

8 - 12 = - 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 = 10, u = - 2

代用を戻す

u = sec (x)

sec (x) = 10, sec (x) = - 2

sec (x) = 10, sec (x) = - 2

sec (x) = 10 : x = arcsec (10) + 2 πn, x = 2 π - arcsec (10) + 2 πn

sec (x) = 10

三角関数の逆数プロパティを適用する

sec (x) = 10

以下の一般解

sec (x) = 10

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

x = arcsec (10) + 2 πn, x = 2 π - arcsec (10) + 2 πn

x = arcsec (10) + 2 πn, x = 2 π - arcsec (10) + 2 πn

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

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

すべての解を組み合わせる

x = arcsec (10) + 2 πn, x = 2 π - arcsec (10) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

10進法形式で解を証明する

x = 1.47062 \dots + 2 πn, x = 2 π - 1.47062 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

グラフ

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