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人気のある三角関数 >

3cot^2(x)-4csc(x)=1

解

3 cot^{2} (x) - 4 csc (x) = 1

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

3 cot^{2} (x) - 4 csc (x) = 1

両辺から

1

を引く

3 cot^{2} (x) - 4 csc (x) - 1 = 0

三角関数の公式を使用して書き換える

- 1 + 3 cot^{2} (x) - 4 csc (x)

ピタゴラスの公式を使用する:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 1 + 3 (csc^{2} (x) - 1) - 4 csc (x)

簡素化

- 1 + 3 (csc^{2} (x) - 1) - 4 csc (x) : 3 csc^{2} (x) - 4 csc (x) - 4

- 1 + 3 (csc^{2} (x) - 1) - 4 csc (x)

拡張

3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 3

3 (csc^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

= 3 csc^{2} (x) - 3 \cdot 1

数を乗じる:

3 \cdot 1 = 3

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

= - 1 + 3 csc^{2} (x) - 3 - 4 csc (x)

簡素化

- 1 + 3 csc^{2} (x) - 3 - 4 csc (x) : 3 csc^{2} (x) - 4 csc (x) - 4

- 1 + 3 csc^{2} (x) - 3 - 4 csc (x)

条件のようなグループ

= 3 csc^{2} (x) - 4 csc (x) - 1 - 3

数を引く:

- 1 - 3 = - 4

= 3 csc^{2} (x) - 4 csc (x) - 4

= 3 csc^{2} (x) - 4 csc (x) - 4

= 3 csc^{2} (x) - 4 csc (x) - 4

- 4 + 3 csc^{2} (x) - 4 csc (x) = 0

置換で解く

- 4 + 3 csc^{2} (x) - 4 csc (x) = 0

仮定:

csc (x) = u

- 4 + 3 u^{2} - 4 u = 0

- 4 + 3 u^{2} - 4 u = 0 : u = 2, u = - \frac{2}{3}

- 4 + 3 u^{2} - 4 u = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 3, b = - 4, c = - 4

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

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

(- 4)^{2} - 4 \cdot 3 (- 4) = 8

(- 4)^{2} - 4 \cdot 3 (- 4)

規則を適用

- (- a) = a

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

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 3 \cdot 4

数を乗じる:

4 \cdot 3 \cdot 4 = 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 3}

解を分離する

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 3}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 3}

u = \frac{- ( - 4 ) + 8}{2 \cdot 3} : 2

\frac{- ( - 4 ) + 8}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{4 + 8}{2 \cdot 3}

数を足す:

4 + 8 = 12

= \frac{12}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 4 ) - 8}{2 \cdot 3} : - \frac{2}{3}

\frac{- ( - 4 ) - 8}{2 \cdot 3}

規則を適用

- (- a) = a

= \frac{4 - 8}{2 \cdot 3}

数を引く:

4 - 8 = - 4

= \frac{- 4}{2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 4}{6}

分数の規則を適用する:

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

= - \frac{4}{6}

共通因数を約分する:

2

= - \frac{2}{3}

二次equationの解:

u = 2, u = - \frac{2}{3}

代用を戻す

u = csc (x)

csc (x) = 2, csc (x) = - \frac{2}{3}

csc (x) = 2, csc (x) = - \frac{2}{3}

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

以下の一般解

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - \frac{2}{3} :

解なし

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

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

解なし

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

人気の例

cos(x)=sin(2x)+cos(3x)

cos (x) = sin (2 x) + cos (3 x)

tan(x)sec(x)-2tan(x)=0

tan (x) sec (x) - 2 tan (x) = 0

-9cos^2(θ)+9=17sin(θ)-8

- 9 cos^{2} (θ) + 9 = 17 sin (θ) - 8

4 tan^{2} (θ) - 9 = 0

(sin(90))/(3.37)=(sin(x))/(0.7)

\frac{sin ( 9 0 ^{\circ} )}{3.37} = \frac{sin ( x )}{0.7}