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人気のある三角関数 >

3cos(x)+1=sec(x)

解

3 cos (x) + 1 = sec (x)

解

x = 2.44587 \dots + 2 πn, x = - 2.44587 \dots + 2 πn, x = 1.12158 \dots + 2 πn, x = 2 π - 1.12158 \dots + 2 πn

+1

度

x = 140.13813 \dots^{\circ} + 36 0^{\circ} n, x = - 140.13813 \dots^{\circ} + 36 0^{\circ} n, x = 64.26187 \dots^{\circ} + 36 0^{\circ} n, x = 295.73812 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 cos (x) + 1 = sec (x)

両辺から

sec (x)

を引く

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

= 1 - sec (x) + 3 \cdot \frac{1}{sec ( x )}

3 \cdot \frac{1}{sec ( x )} = \frac{3}{sec ( x )}

3 \cdot \frac{1}{sec ( x )}

分数を乗じる:

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

= \frac{1 \cdot 3}{sec ( x )}

数を乗じる:

1 \cdot 3 = 3

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

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

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

置換で解く

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

仮定:

sec (x) = u

1 + \frac{3}{u} - u = 0

1 + \frac{3}{u} - u = 0 : u = - \frac{- 1 + 13}{2}, u = \frac{1 + 13}{2}

1 + \frac{3}{u} - u = 0

以下で両辺を乗じる:

u

1 + \frac{3}{u} - u = 0

以下で両辺を乗じる:

u

1 \cdot u + \frac{3}{u} u - uu = 0 \cdot u

簡素化

1 \cdot u + \frac{3}{u} u - uu = 0 \cdot u

簡素化

1 \cdot u : u

1 \cdot u

乗算:

1 \cdot u = u

= u

簡素化

\frac{3}{u} u : 3

\frac{3}{u} u

分数を乗じる:

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

= \frac{3 u}{u}

共通因数を約分する:

u

= 3

簡素化

- uu : - u^{2}

- uu

指数の規則を適用する:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数を足す:

1 + 1 = 2

= - u^{2}

簡素化

0 \cdot u : 0

0 \cdot u

規則を適用

0 \cdot a = 0

= 0

u + 3 - u^{2} = 0

u + 3 - u^{2} = 0

u + 3 - u^{2} = 0

解く

u + 3 - u^{2} = 0 : u = - \frac{- 1 + 13}{2}, u = \frac{1 + 13}{2}

u + 3 - u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

1^{2} - 4 (- 1) \cdot 3 = 13

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

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 3

規則を適用

- (- a) = a

= 1 + 4 \cdot 1 \cdot 3

数を乗じる:

4 \cdot 1 \cdot 3 = 12

= 1 + 12

数を足す:

1 + 12 = 13

= 13

u_{1, 2} = \frac{- 1 \pm 13}{2 ( - 1 )}

解を分離する

u_{1} = \frac{- 1 + 13}{2 ( - 1 )}, u_{2} = \frac{- 1 - 13}{2 ( - 1 )}

u = \frac{- 1 + 13}{2 ( - 1 )} : - \frac{- 1 + 13}{2}

\frac{- 1 + 13}{2 ( - 1 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 13}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 1 + 13}{- 2}

分数の規則を適用する:

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

= - \frac{- 1 + 13}{2}

u = \frac{- 1 - 13}{2 ( - 1 )} : \frac{1 + 13}{2}

\frac{- 1 - 13}{2 ( - 1 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 13}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 1 - 13}{- 2}

分数の規則を適用する:

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

- 1 - 13 = - (1 + 13)

= \frac{1 + 13}{2}

二次equationの解:

u = - \frac{- 1 + 13}{2}, u = \frac{1 + 13}{2}

u = - \frac{- 1 + 13}{2}, u = \frac{1 + 13}{2}

解を検算する

未定義の (特異) 点を求める:

u = 0

1 + \frac{3}{u} - u

の分母をゼロに比較する

u = 0

以下の点は定義されていない

u = 0

未定義のポイントを解に組み合わせる:

u = - \frac{- 1 + 13}{2}, u = \frac{1 + 13}{2}

代用を戻す

u = sec (x)

sec (x) = - \frac{- 1 + 13}{2}, sec (x) = \frac{1 + 13}{2}

sec (x) = - \frac{- 1 + 13}{2}, sec (x) = \frac{1 + 13}{2}

sec (x) = - \frac{- 1 + 13}{2} : x = arcsec (- \frac{- 1 + 13}{2}) + 2 πn, x = - arcsec (- \frac{- 1 + 13}{2}) + 2 πn

sec (x) = - \frac{- 1 + 13}{2}

三角関数の逆数プロパティを適用する

sec (x) = - \frac{- 1 + 13}{2}

以下の一般解

sec (x) = - \frac{- 1 + 13}{2}

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- \frac{- 1 + 13}{2}) + 2 πn, x = - arcsec (- \frac{- 1 + 13}{2}) + 2 πn

x = arcsec (- \frac{- 1 + 13}{2}) + 2 πn, x = - arcsec (- \frac{- 1 + 13}{2}) + 2 πn

sec (x) = \frac{1 + 13}{2} : x = arcsec (\frac{1 + 13}{2}) + 2 πn, x = 2 π - arcsec (\frac{1 + 13}{2}) + 2 πn

sec (x) = \frac{1 + 13}{2}

三角関数の逆数プロパティを適用する

sec (x) = \frac{1 + 13}{2}

以下の一般解

sec (x) = \frac{1 + 13}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (\frac{1 + 13}{2}) + 2 πn, x = 2 π - arcsec (\frac{1 + 13}{2}) + 2 πn

x = arcsec (\frac{1 + 13}{2}) + 2 πn, x = 2 π - arcsec (\frac{1 + 13}{2}) + 2 πn

すべての解を組み合わせる

x = arcsec (- \frac{- 1 + 13}{2}) + 2 πn, x = - arcsec (- \frac{- 1 + 13}{2}) + 2 πn, x = arcsec (\frac{1 + 13}{2}) + 2 πn, x = 2 π - arcsec (\frac{1 + 13}{2}) + 2 πn

10進法形式で解を証明する

x = 2.44587 \dots + 2 πn, x = - 2.44587 \dots + 2 πn, x = 1.12158 \dots + 2 πn, x = 2 π - 1.12158 \dots + 2 πn

グラフ

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