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人気のある三角関数 >

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

解

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

解

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

+1

度

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

sec (x) = u

u^{2} - 4 u = 0

u^{2} - 4 u = 0 : u = 4, u = 0

u^{2} - 4 u = 0

解くとthe二次式

u^{2} - 4 u = 0

二次Equationの公式:

次の場合:

a = 1, b = - 4, c = 0

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

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

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

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

指数の規則を適用する:

n

が偶数であれば

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

(- 4)^{2} = 4^{2}

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

規則を適用

0 \cdot a = 0

= 4^{2} - 0

4^{2} - 0 = 4^{2}

= 4^{2}

累乗根の規則を適用する:

n a^{n} = a,

, 以下を想定

a \geq 0

= 4

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

解を分離する

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

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

\frac{- ( - 4 ) + 4}{2 \cdot 1}

規則を適用

- (- a) = a

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

数を足す:

4 + 4 = 8

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

数を乗じる:

2 \cdot 1 = 2

= \frac{8}{2}

数を割る:

\frac{8}{2} = 4

= 4

u = \frac{- ( - 4 ) - 4}{2 \cdot 1} : 0

\frac{- ( - 4 ) - 4}{2 \cdot 1}

規則を適用

- (- a) = a

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

数を引く:

4 - 4 = 0

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

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{2}

規則を適用

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

= 0

二次equationの解:

u = 4, u = 0

代用を戻す

u = sec (x)

sec (x) = 4, sec (x) = 0

sec (x) = 4, sec (x) = 0

sec (x) = 4 : x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 4

三角関数の逆数プロパティを適用する

sec (x) = 4

以下の一般解

sec (x) = 4

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

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 0 :

解なし

sec (x) = 0

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

解なし

すべての解を組み合わせる

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

10進法形式で解を証明する

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

グラフ

人気の例

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sin (x) = \frac{5}{7}

sec (t) = 2

2cos^2(x)=-cos(x)

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2tan^2(x)+5tan(x)+3=0

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