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人気のある三角関数 >

csc(x)cot(x)=2sqrt(3)

解

csc (x) cot (x) = 23

解

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

解答ステップ

csc (x) cot (x) = 23

両辺から

23

を引く

csc (x) cot (x) - 23 = 0

サイン, コサインで表わす

- 23 + cot (x) csc (x)

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 23 + \frac{cos ( x )}{sin ( x )} csc (x)

基本的な三角関数の公式を使用する:

csc (x) = \frac{1}{sin ( x )}

= - 23 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

簡素化

- 23 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : \frac{- 2 3 sin ^{2} ( x ) + cos ( x )}{sin ^{2} ( x )}

- 23 + \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} = \frac{cos ( x )}{sin ^{2} ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

分数を乗じる:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{cos ( x ) \cdot 1}{sin ( x ) sin ( x )}

乗算:

cos (x) \cdot 1 = cos (x)

= \frac{cos ( x )}{sin ( x ) sin ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= sin^{2} (x)

= \frac{cos ( x )}{sin ^{2} ( x )}

= - 23 + \frac{cos ( x )}{sin ^{2} ( x )}

元を分数に変換する:

23 = \frac{2 \cdot 3 sin ^{2} ( x )}{sin ^{2} ( x )}

= - \frac{2 3 sin ^{2} ( x )}{sin ^{2} ( x )} + \frac{cos ( x )}{sin ^{2} ( x )}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 3 sin ^{2} ( x ) + cos ( x )}{sin ^{2} ( x )}

= \frac{- 2 3 sin ^{2} ( x ) + cos ( x )}{sin ^{2} ( x )}

\frac{cos ( x ) - 2 sin ^{2} ( x ) 3}{sin ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (x) - 2 sin^{2} (x) 3 = 0

三角関数の公式を使用して書き換える

cos (x) - 2 sin^{2} (x) 3

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= cos (x) - 2 (1 - cos^{2} (x)) 3

cos (x) - (1 - cos^{2} (x)) \cdot 23 = 0

置換で解く

cos (x) - (1 - cos^{2} (x)) \cdot 23 = 0

仮定:

cos (x) = u

u - (1 - u^{2}) \cdot 23 = 0

u - (1 - u^{2}) \cdot 23 = 0 : u = \frac{3}{2}, u = - \frac{2 3}{3}

u - (1 - u^{2}) \cdot 23 = 0

拡張

u - (1 - u^{2}) \cdot 23 : u - 23 + 23 u^{2}

u - (1 - u^{2}) \cdot 23

= u - 23 (1 - u^{2})

拡張

- 23 (1 - u^{2}) : - 23 + 23 u^{2}

- 23 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = - 23, b = 1, c = u^{2}

= - 23 \cdot 1 - (- 23) u^{2}

マイナス・プラスの規則を適用する

- (- a) = a

= - 2 \cdot 1 \cdot 3 + 23 u^{2}

数を乗じる:

2 \cdot 1 = 2

= - 23 + 23 u^{2}

= u - 23 + 23 u^{2}

u - 23 + 23 u^{2} = 0

標準的な形式で書く

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

23 u^{2} + u - 23 = 0

解くとthe二次式

23 u^{2} + u - 23 = 0

二次Equationの公式:

次の場合:

a = 23, b = 1, c = - 23

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

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

1^{2} - 4 \cdot 23 (- 23) = 7

1^{2} - 4 \cdot 23 (- 23)

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 23 (- 23)

規則を適用

- (- a) = a

= 1 + 4 \cdot 23 \cdot 23

4 \cdot 23 \cdot 23 = 48

4 \cdot 23 \cdot 23

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 1633

累乗根の規則を適用する:

a a = a

33 = 3

= 16 \cdot 3

数を乗じる:

16 \cdot 3 = 48

= 48

= 1 + 48

数を足す:

1 + 48 = 49

= 49

数を因数に分解する:

49 = 7^{2}

= 7^{2}

累乗根の規則を適用する:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 1 \pm 7}{2 \cdot 2 3}

解を分離する

u_{1} = \frac{- 1 + 7}{2 \cdot 2 3}, u_{2} = \frac{- 1 - 7}{2 \cdot 2 3}

u = \frac{- 1 + 7}{2 \cdot 2 3} : \frac{3}{2}

\frac{- 1 + 7}{2 \cdot 2 3}

数を足す/引く:

- 1 + 7 = 6

= \frac{6}{2 \cdot 2 3}

数を乗じる:

2 \cdot 2 = 4

= \frac{6}{4 3}

共通因数を約分する:

2

= \frac{3}{2 3}

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

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

指数の規則を適用する:

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

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

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

数を引く:

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

= \frac{3 ^{\frac{1}{2}}}{2}

累乗根の規則を適用する:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= \frac{3}{2}

u = \frac{- 1 - 7}{2 \cdot 2 3} : - \frac{2 3}{3}

\frac{- 1 - 7}{2 \cdot 2 3}

数を引く:

- 1 - 7 = - 8

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 8}{4 3}

分数の規則を適用する:

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

= - \frac{8}{4 3}

数を割る:

\frac{8}{4} = 2

= - \frac{2}{3}

有理化する

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

共役で乗じる

\frac{3}{3}

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

33 = 3

33

累乗根の規則を適用する:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

以下の一般解

cos (x) = \frac{3}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

解なし

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

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

グラフ

人気の例

cos^2(θ)+2sin(θ)+1=0

cos^{2} (θ) + 2 sin (θ) + 1 = 0

arcsec(x)=arcsec(2)

arcsec (x) = arcsec (2)

tan^2(x)+(sqrt(3)-1)tan(x)-sqrt(3)=0

tan^{2} (x) + (3 - 1) tan (x) - 3 = 0

2 sec (x) - 5 = 0

cos (x) = 0.84