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人気のある三角関数 >

sec(x)=3cos(x)-tan(x)

解

sec (x) = 3 cos (x) - tan (x)

解

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sec (x) = 3 cos (x) - tan (x)

両辺から

3 cos (x) - tan (x)

を引く

sec (x) - 3 cos (x) + tan (x) = 0

サイン, コサインで表わす

\frac{1}{cos ( x )} - 3 cos (x) + \frac{sin ( x )}{cos ( x )} = 0

簡素化

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

\frac{1}{cos ( x )} - 3 cos (x) + \frac{sin ( x )}{cos ( x )}

分数を組み合わせる

\frac{1}{cos ( x )} + \frac{sin ( x )}{cos ( x )} : \frac{1 + sin ( x )}{cos ( x )}

規則を適用

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

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

= \frac{sin ( x ) + 1}{cos ( x )} - 3 cos (x)

元を分数に変換する:

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

= \frac{1 + sin ( x )}{cos ( x )} - \frac{3 cos ( 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{1 + sin ( x ) - 3 cos ( x ) cos ( x )}{cos ( x )}

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

1 + sin (x) - 3 cos (x) cos (x)

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

3 cos (x) cos (x)

指数の規則を適用する:

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

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

= 3 cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 3 cos^{2} (x)

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

= \frac{1 + sin ( x ) - 3 cos ^{2} ( x )}{cos ( x )}

\frac{1 + sin ( x ) - 3 cos ^{2} ( x )}{cos ( x )} = 0

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

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

両辺に

3 cos^{2} (x)

を足す

1 + sin (x) = 3 cos^{2} (x)

両辺を2乗する

(1 + sin (x))^{2} = (3 cos^{2} (x))^{2}

両辺から

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

を引く

(1 + sin (x))^{2} - 9 cos^{4} (x) = 0

因数

(1 + sin (x))^{2} - 9 cos^{4} (x) : (1 + sin (x) + 3 cos^{2} (x)) (1 + sin (x) - 3 cos^{2} (x))

(1 + sin (x))^{2} - 9 cos^{4} (x)

(1 + sin (x))^{2} - 9 cos^{4} (x)

を書き換え

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

(1 + sin (x))^{2} - 9 cos^{4} (x)

9

を書き換え

3^{2}

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

指数の規則を適用する:

a^{b c} = (a^{b})^{c}

cos^{4} (x) = (cos^{2} (x))^{2}

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

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

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

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

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

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

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

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

改良

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

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

各部分を別個に解く

1 + sin (x) + 3 cos^{2} (x) = 0 or 1 + sin (x) - 3 cos^{2} (x) = 0

1 + sin (x) + 3 cos^{2} (x) = 0 : x = \frac{3 π}{2} + 2 πn

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

三角関数の公式を使用して書き換える

1 + sin (x) + 3 cos^{2} (x)

ピタゴラスの公式を使用する:

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

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

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

簡素化

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

1 + sin (x) + 3 (1 - sin^{2} (x))

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

数を乗じる:

3 \cdot 1 = 3

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

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

簡素化

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

1 + sin (x) + 3 - 3 sin^{2} (x)

条件のようなグループ

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

数を足す:

1 + 3 = 4

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

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

4 + u - 3 u^{2} = 0 : u = - 1, u = \frac{4}{3}

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 3, b = 1, c = 4

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

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

1^{2} - 4 (- 3) \cdot 4 = 7

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 3) \cdot 4

規則を適用

- (- a) = a

= 1 + 4 \cdot 3 \cdot 4

数を乗じる:

4 \cdot 3 \cdot 4 = 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 ( - 3 )}

解を分離する

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

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

\frac{- 1 + 7}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

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

数を足す/引く:

- 1 + 7 = 6

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

数を乗じる:

2 \cdot 3 = 6

= \frac{6}{- 6}

分数の規則を適用する:

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

= - \frac{6}{6}

規則を適用

\frac{a}{a} = 1

= - 1

u = \frac{- 1 - 7}{2 ( - 3 )} : \frac{4}{3}

\frac{- 1 - 7}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

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

数を引く:

- 1 - 7 = - 8

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

数を乗じる:

2 \cdot 3 = 6

= \frac{- 8}{- 6}

分数の規則を適用する:

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

= \frac{8}{6}

共通因数を約分する:

2

= \frac{4}{3}

二次equationの解:

u = - 1, u = \frac{4}{3}

代用を戻す

u = sin (x)

sin (x) = - 1, sin (x) = \frac{4}{3}

sin (x) = - 1, sin (x) = \frac{4}{3}

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

以下の一般解

sin (x) = - 1

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

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = \frac{4}{3} :

解なし

sin (x) = \frac{4}{3}

- 1 \leq sin (x) \leq 1

解なし

すべての解を組み合わせる

x = \frac{3 π}{2} + 2 πn

1 + sin (x) - 3 cos^{2} (x) = 0 : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn, x = \frac{3 π}{2} + 2 πn

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

三角関数の公式を使用して書き換える

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

ピタゴラスの公式を使用する:

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

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

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

簡素化

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

1 + sin (x) - 3 (1 - sin^{2} (x))

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

マイナス・プラスの規則を適用する

- (- a) = a

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

数を乗じる:

3 \cdot 1 = 3

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

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

簡素化

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

1 + sin (x) - 3 + 3 sin^{2} (x)

条件のようなグループ

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

数を足す/引く:

1 - 3 = - 2

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

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

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

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

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 3, b = 1, c = - 2

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

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

1^{2} - 4 \cdot 3 (- 2) = 5

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 3 (- 2)

規則を適用

- (- a) = a

= 1 + 4 \cdot 3 \cdot 2

数を乗じる:

4 \cdot 3 \cdot 2 = 24

= 1 + 24

数を足す:

1 + 24 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 \cdot 3}

解を分離する

u_{1} = \frac{- 1 + 5}{2 \cdot 3}, u_{2} = \frac{- 1 - 5}{2 \cdot 3}

u = \frac{- 1 + 5}{2 \cdot 3} : \frac{2}{3}

\frac{- 1 + 5}{2 \cdot 3}

数を足す/引く:

- 1 + 5 = 4

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

数を乗じる:

2 \cdot 3 = 6

= \frac{4}{6}

共通因数を約分する:

2

= \frac{2}{3}

u = \frac{- 1 - 5}{2 \cdot 3} : - 1

\frac{- 1 - 5}{2 \cdot 3}

数を引く:

- 1 - 5 = - 6

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

数を乗じる:

2 \cdot 3 = 6

= \frac{- 6}{6}

分数の規則を適用する:

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

= - \frac{6}{6}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = sin (x)

sin (x) = \frac{2}{3}, sin (x) = - 1

sin (x) = \frac{2}{3}, sin (x) = - 1

sin (x) = \frac{2}{3} : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

以下の一般解

sin (x) = - 1

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

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn, x = \frac{3 π}{2} + 2 πn

すべての解を組み合わせる

x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

元のequationに当てはめて解を検算する

sec (x) = 3 cos (x) - 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

\frac{3 π}{2} + 2 πn :

真

\frac{3 π}{2} + 2 πn

挿入

n = 1

\frac{3 π}{2} + 2 π 1

sec (x) = 3 cos (x) - tan (x)

の挿入向け

x = \frac{3 π}{2} + 2 π 1

sec (\frac{3 π}{2} + 2 π 1) = 3 cos (\frac{3 π}{2} + 2 π 1) - tan (\frac{3 π}{2} + 2 π 1)

改良

- \infty = - \infty

\Rightarrow 真

解答を確認する

arcsin (\frac{2}{3}) + 2 πn :

真

arcsin (\frac{2}{3}) + 2 πn

挿入

n = 1

arcsin (\frac{2}{3}) + 2 π 1

sec (x) = 3 cos (x) - tan (x)

の挿入向け

x = arcsin (\frac{2}{3}) + 2 π 1

sec (arcsin (\frac{2}{3}) + 2 π 1) = 3 cos (arcsin (\frac{2}{3}) + 2 π 1) - tan (arcsin (\frac{2}{3}) + 2 π 1)

改良

1.34164 \dots = 1.34164 \dots

\Rightarrow 真

解答を確認する

π - arcsin (\frac{2}{3}) + 2 πn :

真

π - arcsin (\frac{2}{3}) + 2 πn

挿入

n = 1

π - arcsin (\frac{2}{3}) + 2 π 1

sec (x) = 3 cos (x) - tan (x)

の挿入向け

x = π - arcsin (\frac{2}{3}) + 2 π 1

sec (π - arcsin (\frac{2}{3}) + 2 π 1) = 3 cos (π - arcsin (\frac{2}{3}) + 2 π 1) - tan (π - arcsin (\frac{2}{3}) + 2 π 1)

改良

- 1.34164 \dots = - 1.34164 \dots

\Rightarrow 真

x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

equationは以下で未定義のため:

\frac{3 π}{2} + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

10進法形式で解を証明する

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

人気の例

arctan(x^2-2x)=0

arctan (x^{2} - 2 x) = 0

sin(x)=(6.57*10^{-7})/(1.8*10^{-6)}

sin (x^{\circ}) = \frac{6.57 \cdot 1 0 ^{- 7}}{1.8 \cdot 1 0 ^{- 6}}

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

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

solvefor y,xtan(y)-x^2sin(x)+3x^3=0

so l v e f or y, x tan (y) - x^{2} sin (x) + 3 x^{3} = 0

(((1-sin^2(x)))/((1+2*sin^2(x))))=0.45

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.45