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人気のある三角関数 >

(cot(x)+1/(cot(x)))/(cos(x))=sec(x)

解

\frac{cot ( x ) + \frac{1}{c o t ( x )}}{cos ( x )} = sec (x)

解

以下の解はない : x \in R

解答ステップ

\frac{cot ( x ) + \frac{1}{c o t ( x )}}{cos ( x )} = sec (x)

両辺から

sec (x)

を引く

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x) = 0

簡素化

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x) : \frac{cot ^{2} ( x ) + 1 - sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( x )}

\frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - sec (x)

元を分数に変換する:

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

= \frac{cot ^{2} ( x ) + 1}{cot ( x ) cos ( x )} - \frac{sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( 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{cot ^{2} ( x ) + 1 - sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( x )}

\frac{cot ^{2} ( x ) + 1 - sec ( x ) cot ( x ) cos ( x )}{cot ( x ) cos ( x )} = 0

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

cot^{2} (x) + 1 - sec (x) cot (x) cos (x) = 0

三角関数の公式を使用して書き換える

1 + cot^{2} (x) - cos (x) cot (x) sec (x)

基本的な三角関数の公式を使用する:

cos (x) = \frac{1}{sec ( x )}

= 1 + cot^{2} (x) - \frac{1}{sec ( x )} cot (x) sec (x)

\frac{1}{sec ( x )} cot (x) sec (x) = cot (x)

\frac{1}{sec ( x )} cot (x) sec (x)

分数を乗じる:

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

= \frac{1 \cdot cot ( x ) sec ( x )}{sec ( x )}

共通因数を約分する:

sec (x)

= 1 \cdot cot (x)

乗算:

1 \cdot cot (x) = cot (x)

= cot (x)

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

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

置換で解く

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

仮定:

cot (x) = u

1 - u + u^{2} = 0

1 - u + u^{2} = 0 : u = \frac{1}{2} + i \frac{3}{2}, u = \frac{1}{2} - i \frac{3}{2}

1 - u + u^{2} = 0

標準的な形式で書く

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

u^{2} - u + 1 = 0

解くとthe二次式

u^{2} - u + 1 = 0

二次Equationの公式:

次の場合:

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

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

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

簡素化

(- 1)^{2} - 4 \cdot 1 \cdot 1 : 3 i

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

(- 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 1 = 4

4 \cdot 1 \cdot 1

数を乗じる:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 - 4

数を引く:

1 - 4 = - 3

= - 3

累乗根の規則を適用する:

- a = - 1 a

- 3 = - 1 3

= - 1 3

虚数の規則を適用する:

- 1 = i

= 3 i

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

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を乗じる:

2 \cdot 1 = 2

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

標準的な複素数形式で

\frac{1 + 3 i}{2}

を書き換える:

\frac{1}{2} + \frac{3}{2} i

\frac{1 + 3 i}{2}

分数の規則を適用する:

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

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

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

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

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

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

規則を適用

- (- a) = a

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

数を乗じる:

2 \cdot 1 = 2

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

標準的な複素数形式で

\frac{1 - 3 i}{2}

を書き換える:

\frac{1}{2} - \frac{3}{2} i

\frac{1 - 3 i}{2}

分数の規則を適用する:

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

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

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

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

二次equationの解:

u = \frac{1}{2} + i \frac{3}{2}, u = \frac{1}{2} - i \frac{3}{2}

代用を戻す

u = cot (x)

cot (x) = \frac{1}{2} + i \frac{3}{2}, cot (x) = \frac{1}{2} - i \frac{3}{2}

cot (x) = \frac{1}{2} + i \frac{3}{2}, cot (x) = \frac{1}{2} - i \frac{3}{2}

cot (x) = \frac{1}{2} + i \frac{3}{2} :

解なし

cot (x) = \frac{1}{2} + i \frac{3}{2}

解なし

cot (x) = \frac{1}{2} - i \frac{3}{2} :

解なし

cot (x) = \frac{1}{2} - i \frac{3}{2}

解なし

すべての解を組み合わせる

以下の解はない : x \in R

グラフ

人気の例

sec(θ)=2,(tan(θ)-sin(θ))^2+(1-cos(θ))^2

sec (θ) = 2, (tan (θ) - sin (θ))^{2} + (1 - cos (θ))^{2}

(1+csc(A))(1-csc(A))=-cot(A)

(1 + csc (A)) (1 - csc (A)) = - cot (A)

6cos^2(3x)-cos(3x)-2=0

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

cos(2x)-3cos(x)+1=0

cos (2 x) - 3 cos (x) + 1 = 0

2cos^2(x)-cos(x)-1=0,0<= x<= 360

2 cos^{2} (x) - cos (x) - 1 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}