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人気のある三角関数 >

4/(cos(x))-6=tan(x)

解

\frac{4}{cos ( x )} - 6 = tan (x)

解

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

+1

度

x = 320.57910 \dots^{\circ} + 36 0^{\circ} n, x = 58.34553 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

\frac{4}{cos ( x )} - 6 = tan (x)

両辺から

tan (x)

を引く

\frac{4}{cos ( x )} - 6 - tan (x) = 0

簡素化

\frac{4}{cos ( x )} - 6 - tan (x) : \frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4}{cos ( x )} - 6 - tan (x)

元を分数に変換する:

6 = \frac{6 cos ( x )}{cos ( x )}, tan (x) = \frac{tan ( x ) cos ( x )}{cos ( x )}

= \frac{4}{cos ( x )} - \frac{6 cos ( x )}{cos ( x )} - \frac{tan ( x ) cos ( 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{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )}

\frac{4 - 6 cos ( x ) - tan ( x ) cos ( x )}{cos ( x )} = 0

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

4 - 6 cos (x) - tan (x) cos (x) = 0

サイン, コサインで表わす

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) = 0

簡素化

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x) : 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - \frac{sin ( x )}{cos ( x )} cos (x)

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

= sin (x)

= 4 - 6 cos (x) - sin (x)

4 - 6 cos (x) - sin (x) = 0

両辺に

sin (x)

を足す

4 - 6 cos (x) = sin (x)

両辺を2乗する

(4 - 6 cos (x))^{2} = sin^{2} (x)

両辺から

sin^{2} (x)

を引く

(4 - 6 cos (x))^{2} - sin^{2} (x) = 0

三角関数の公式を使用して書き換える

(4 - 6 cos (x))^{2} - sin^{2} (x)

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)

= (4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

簡素化

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x)) : 37 cos^{2} (x) - 48 cos (x) + 15

(4 - 6 cos (x))^{2} - (1 - cos^{2} (x))

(4 - 6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

完全平方式を適用する:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 4, b = 6 cos (x)

= 4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

簡素化

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2} : 16 - 48 cos (x) + 36 cos^{2} (x)

4^{2} - 2 \cdot 4 \cdot 6 cos (x) + (6 cos (x))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 4 \cdot 6 cos (x) = 48 cos (x)

2 \cdot 4 \cdot 6 cos (x)

数を乗じる:

2 \cdot 4 \cdot 6 = 48

= 48 cos (x)

(6 cos (x))^{2} = 36 cos^{2} (x)

(6 cos (x))^{2}

指数の規則を適用する:

(a \cdot b)^{n} = a^{n} b^{n}

= 6^{2} cos^{2} (x)

6^{2} = 36

= 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x)

= 16 - 48 cos (x) + 36 cos^{2} (x) - (1 - cos^{2} (x))

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

括弧を分配する

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

マイナス・プラスの規則を適用する

- (- a) = a, - (a) = - a

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

= 16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

簡素化

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x) : 37 cos^{2} (x) - 48 cos (x) + 15

16 - 48 cos (x) + 36 cos^{2} (x) - 1 + cos^{2} (x)

条件のようなグループ

= - 48 cos (x) + 36 cos^{2} (x) + cos^{2} (x) + 16 - 1

類似した元を足す:

36 cos^{2} (x) + cos^{2} (x) = 37 cos^{2} (x)

= - 48 cos (x) + 37 cos^{2} (x) + 16 - 1

数を足す/引く:

16 - 1 = 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

= 37 cos^{2} (x) - 48 cos (x) + 15

15 + 37 cos^{2} (x) - 48 cos (x) = 0

置換で解く

15 + 37 cos^{2} (x) - 48 cos (x) = 0

仮定:

cos (x) = u

15 + 37 u^{2} - 48 u = 0

15 + 37 u^{2} - 48 u = 0 : u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

15 + 37 u^{2} - 48 u = 0

標準的な形式で書く

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

37 u^{2} - 48 u + 15 = 0

解くとthe二次式

37 u^{2} - 48 u + 15 = 0

二次Equationの公式:

次の場合:

a = 37, b = - 48, c = 15

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

u_{1, 2} = \frac{- ( - 48 ) \pm ( - 48 ) ^{2} - 4 \cdot 37 \cdot 15}{2 \cdot 37}

(- 48)^{2} - 4 \cdot 37 \cdot 15 = 221

(- 48)^{2} - 4 \cdot 37 \cdot 15

指数の規則を適用する:

n

が偶数であれば

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

(- 48)^{2} = 4 8^{2}

= 4 8^{2} - 4 \cdot 37 \cdot 15

数を乗じる:

4 \cdot 37 \cdot 15 = 2220

= 4 8^{2} - 2220

4 8^{2} = 2304

= 2304 - 2220

数を引く:

2304 - 2220 = 84

= 84

以下の素因数分解:

84 : 2^{2} \cdot 3 \cdot 7

84

84284 = 42 \cdot 2

で割る

= 2 \cdot 42

42242 = 21 \cdot 2

で割る

= 2 \cdot 2 \cdot 21

21321 = 7 \cdot 3

で割る

= 2 \cdot 2 \cdot 3 \cdot 7

2, 3, 7

はすべて素数である。ゆえにさらに因数分解することはできない

= 2 \cdot 2 \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

= 2^{2} \cdot 3 \cdot 7

累乗根の規則を適用する:

n ab = n a n b

= 2^{2} 3 \cdot 7

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 7

改良

= 221

u_{1, 2} = \frac{- ( - 48 ) \pm 2 21}{2 \cdot 37}

解を分離する

u_{1} = \frac{- ( - 48 ) + 2 21}{2 \cdot 37}, u_{2} = \frac{- ( - 48 ) - 2 21}{2 \cdot 37}

u = \frac{- ( - 48 ) + 2 21}{2 \cdot 37} : \frac{24 + 21}{37}

\frac{- ( - 48 ) + 2 21}{2 \cdot 37}

規則を適用

- (- a) = a

= \frac{48 + 2 21}{2 \cdot 37}

数を乗じる:

2 \cdot 37 = 74

= \frac{48 + 2 21}{74}

因数

48 + 221 : 2 (24 + 21)

48 + 221

書き換え

= 2 \cdot 24 + 221

共通項をくくり出す

2

= 2 (24 + 21)

= \frac{2 ( 24 + 21 )}{74}

共通因数を約分する:

2

= \frac{24 + 21}{37}

u = \frac{- ( - 48 ) - 2 21}{2 \cdot 37} : \frac{24 - 21}{37}

\frac{- ( - 48 ) - 2 21}{2 \cdot 37}

規則を適用

- (- a) = a

= \frac{48 - 2 21}{2 \cdot 37}

数を乗じる:

2 \cdot 37 = 74

= \frac{48 - 2 21}{74}

因数

48 - 221 : 2 (24 - 21)

48 - 221

書き換え

= 2 \cdot 24 - 221

共通項をくくり出す

2

= 2 (24 - 21)

= \frac{2 ( 24 - 21 )}{74}

共通因数を約分する:

2

= \frac{24 - 21}{37}

二次equationの解:

u = \frac{24 + 21}{37}, u = \frac{24 - 21}{37}

代用を戻す

u = cos (x)

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37}, cos (x) = \frac{24 - 21}{37}

cos (x) = \frac{24 + 21}{37} : x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 + 21}{37}

三角関数の逆数プロパティを適用する

cos (x) = \frac{24 + 21}{37}

以下の一般解

cos (x) = \frac{24 + 21}{37}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37} : x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

cos (x) = \frac{24 - 21}{37}

三角関数の逆数プロパティを適用する

cos (x) = \frac{24 - 21}{37}

以下の一般解

cos (x) = \frac{24 - 21}{37}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{24 + 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn, x = 2 π - arccos (\frac{24 - 21}{37}) + 2 πn

元のequationに当てはめて解を検算する

\frac{4}{cos ( x )} - 6 = tan (x)

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

arccos (\frac{24 + 21}{37}) + 2 πn :

偽

arccos (\frac{24 + 21}{37}) + 2 πn

挿入

n = 1

arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( x )} - 6 = tan (x)

の挿入向け

x = arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 + 21}{37}) + 2 π 1)

改良

- 0.82202 \dots = 0.82202 \dots

\Rightarrow 偽

解答を確認する

2 π - arccos (\frac{24 + 21}{37}) + 2 πn :

真

2 π - arccos (\frac{24 + 21}{37}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( x )} - 6 = tan (x)

の挿入向け

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 + 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 + 21}{37}) + 2 π 1)

改良

- 0.82202 \dots = - 0.82202 \dots

\Rightarrow 真

解答を確認する

arccos (\frac{24 - 21}{37}) + 2 πn :

真

arccos (\frac{24 - 21}{37}) + 2 πn

挿入

n = 1

arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( x )} - 6 = tan (x)

の挿入向け

x = arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (arccos (\frac{24 - 21}{37}) + 2 π 1)

改良

1.62202 \dots = 1.62202 \dots

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{24 - 21}{37}) + 2 πn :

偽

2 π - arccos (\frac{24 - 21}{37}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( x )} - 6 = tan (x)

の挿入向け

x = 2 π - arccos (\frac{24 - 21}{37}) + 2 π 1

\frac{4}{cos ( 2 π - arccos ( \frac{24 - 21}{37} ) + 2 π 1 )} - 6 = tan (2 π - arccos (\frac{24 - 21}{37}) + 2 π 1)

改良

1.62202 \dots = - 1.62202 \dots

\Rightarrow 偽

x = 2 π - arccos (\frac{24 + 21}{37}) + 2 πn, x = arccos (\frac{24 - 21}{37}) + 2 πn

10進法形式で解を証明する

x = 2 π - 0.68802 \dots + 2 πn, x = 1.01832 \dots + 2 πn

グラフ

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