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人気のある三角関数 >

sec^2(x)-3sec(x)=0

解

sec^{2} (x) - 3 sec (x) = 0

解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

度

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sec^{2} (x) - 3 sec (x) = 0

置換で解く

sec^{2} (x) - 3 sec (x) = 0

仮定:

sec (x) = u

u^{2} - 3 u = 0

u^{2} - 3 u = 0 : u = 3, u = 0

u^{2} - 3 u = 0

解くとthe二次式

u^{2} - 3 u = 0

二次Equationの公式:

次の場合:

a = 1, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot 0 = 3

(- 3)^{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}

(- 3)^{2} = 3^{2}

= 3^{2} - 4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 3^{2} - 0

3^{2} - 0 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a,

, 以下を想定

a \geq 0

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 1}

u = \frac{- ( - 3 ) + 3}{2 \cdot 1} : 3

\frac{- ( - 3 ) + 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{3 + 3}{2 \cdot 1}

数を足す:

3 + 3 = 6

= \frac{6}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{6}{2}

数を割る:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 3 ) - 3}{2 \cdot 1} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{3 - 3}{2 \cdot 1}

数を引く:

3 - 3 = 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 = 3, u = 0

代用を戻す

u = sec (x)

sec (x) = 3, sec (x) = 0

sec (x) = 3, sec (x) = 0

sec (x) = 3 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = 3

三角関数の逆数プロパティを適用する

sec (x) = 3

以下の一般解

sec (x) = 3

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 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 (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

10進法形式で解を証明する

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

グラフ

人気の例

sqrt(3)cot(x)+1=0

3 cot (x) + 1 = 0

tan (x) = - \frac{2}{3}

tan (θ) = \frac{5}{3}

tan (x) = \frac{3}{5}

6sec^2(θ)tan(θ)=12tan(θ)

6 sec^{2} (θ) tan (θ) = 12 tan (θ)