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人気のある三角関数 >

csc^2(x)-csc(x)-20=0

解

csc^{2} (x) - csc (x) - 20 = 0

解

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

+1

度

x = 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 168.46304 \dots^{\circ} + 36 0^{\circ} n, x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

csc^{2} (x) - csc (x) - 20 = 0

置換で解く

csc^{2} (x) - csc (x) - 20 = 0

仮定:

csc (x) = u

u^{2} - u - 20 = 0

u^{2} - u - 20 = 0 : u = 5, u = - 4

u^{2} - u - 20 = 0

解くとthe二次式

u^{2} - u - 20 = 0

二次Equationの公式:

次の場合:

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

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

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

(- 1)^{2} - 4 \cdot 1 \cdot (- 20) = 9

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

規則を適用

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

指数の規則を適用する:

n

が偶数であれば

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

(- 1)^{2} = 1^{2}

= 1^{2}

規則を適用

1^{a} = 1

= 1

4 \cdot 1 \cdot 20 = 80

4 \cdot 1 \cdot 20

数を乗じる:

4 \cdot 1 \cdot 20 = 80

= 80

= 1 + 80

数を足す:

1 + 80 = 81

= 81

数を因数に分解する:

81 = 9^{2}

= 9^{2}

累乗根の規則を適用する:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 1 ) \pm 9}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 1 ) + 9}{2 \cdot 1}, u_{2} = \frac{- ( - 1 ) - 9}{2 \cdot 1}

u = \frac{- ( - 1 ) + 9}{2 \cdot 1} : 5

\frac{- ( - 1 ) + 9}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{1 + 9}{2 \cdot 1}

数を足す:

1 + 9 = 10

= \frac{10}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{10}{2}

数を割る:

\frac{10}{2} = 5

= 5

u = \frac{- ( - 1 ) - 9}{2 \cdot 1} : - 4

\frac{- ( - 1 ) - 9}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{1 - 9}{2 \cdot 1}

数を引く:

1 - 9 = - 8

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

数を乗じる:

2 \cdot 1 = 2

= \frac{- 8}{2}

分数の規則を適用する:

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

= - \frac{8}{2}

数を割る:

\frac{8}{2} = 4

= - 4

二次equationの解:

u = 5, u = - 4

代用を戻す

u = csc (x)

csc (x) = 5, csc (x) = - 4

csc (x) = 5, csc (x) = - 4

csc (x) = 5 : x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 5

三角関数の逆数プロパティを適用する

csc (x) = 5

以下の一般解

csc (x) = 5

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = - 4 : x = arccsc (- 4) + 2 πn, x = π + arccsc (4) + 2 πn

csc (x) = - 4

三角関数の逆数プロパティを適用する

csc (x) = - 4

以下の一般解

csc (x) = - 4

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

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

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

すべての解を組み合わせる

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn, x = arccsc (- 4) + 2 πn, x = π + arccsc (4) + 2 πn

10進法形式で解を証明する

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

グラフ

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