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人気のある三角関数 >

5sin^2(x)-cos(x)=2

解

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

解

x = 2.64882 \dots + 2 πn, x = - 2.64882 \dots + 2 πn, x = 0.82163 \dots + 2 πn, x = 2 π - 0.82163 \dots + 2 πn

+1

度

x = 151.76625 \dots^{\circ} + 36 0^{\circ} n, x = - 151.76625 \dots^{\circ} + 36 0^{\circ} n, x = 47.07621 \dots^{\circ} + 36 0^{\circ} n, x = 312.92378 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

2

を引く

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

三角関数の公式を使用して書き換える

- 2 - cos (x) + 5 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)

= - 2 - cos (x) + 5 (1 - cos^{2} (x))

簡素化

- 2 - cos (x) + 5 (1 - cos^{2} (x)) : - 5 cos^{2} (x) - cos (x) + 3

- 2 - cos (x) + 5 (1 - cos^{2} (x))

拡張

5 (1 - cos^{2} (x)) : 5 - 5 cos^{2} (x)

5 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 5, b = 1, c = cos^{2} (x)

= 5 \cdot 1 - 5 cos^{2} (x)

数を乗じる:

5 \cdot 1 = 5

= 5 - 5 cos^{2} (x)

= - 2 - cos (x) + 5 - 5 cos^{2} (x)

簡素化

- 2 - cos (x) + 5 - 5 cos^{2} (x) : - 5 cos^{2} (x) - cos (x) + 3

- 2 - cos (x) + 5 - 5 cos^{2} (x)

条件のようなグループ

= - cos (x) - 5 cos^{2} (x) - 2 + 5

数を足す/引く:

- 2 + 5 = 3

= - 5 cos^{2} (x) - cos (x) + 3

= - 5 cos^{2} (x) - cos (x) + 3

= - 5 cos^{2} (x) - cos (x) + 3

3 - cos (x) - 5 cos^{2} (x) = 0

置換で解く

3 - cos (x) - 5 cos^{2} (x) = 0

仮定:

cos (x) = u

3 - u - 5 u^{2} = 0

3 - u - 5 u^{2} = 0 : u = - \frac{1 + 61}{10}, u = \frac{61 - 1}{10}

3 - u - 5 u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 5, b = - 1, c = 3

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 5 ) \cdot 3}{2 ( - 5 )}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 5 ) \cdot 3}{2 ( - 5 )}

(- 1)^{2} - 4 (- 5) \cdot 3 = 61

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

規則を適用

- (- a) = a

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

(- 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 5 \cdot 3 = 60

4 \cdot 5 \cdot 3

数を乗じる:

4 \cdot 5 \cdot 3 = 60

= 60

= 1 + 60

数を足す:

1 + 60 = 61

= 61

u_{1, 2} = \frac{- ( - 1 ) \pm 61}{2 ( - 5 )}

解を分離する

u_{1} = \frac{- ( - 1 ) + 61}{2 ( - 5 )}, u_{2} = \frac{- ( - 1 ) - 61}{2 ( - 5 )}

u = \frac{- ( - 1 ) + 61}{2 ( - 5 )} : - \frac{1 + 61}{10}

\frac{- ( - 1 ) + 61}{2 ( - 5 )}

括弧を削除する:

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

= \frac{1 + 61}{- 2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{1 + 61}{- 10}

分数の規則を適用する:

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

= - \frac{1 + 61}{10}

u = \frac{- ( - 1 ) - 61}{2 ( - 5 )} : \frac{61 - 1}{10}

\frac{- ( - 1 ) - 61}{2 ( - 5 )}

括弧を削除する:

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

= \frac{1 - 61}{- 2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{1 - 61}{- 10}

分数の規則を適用する:

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

1 - 61 = - (61 - 1)

= \frac{61 - 1}{10}

二次equationの解:

u = - \frac{1 + 61}{10}, u = \frac{61 - 1}{10}

代用を戻す

u = cos (x)

cos (x) = - \frac{1 + 61}{10}, cos (x) = \frac{61 - 1}{10}

cos (x) = - \frac{1 + 61}{10}, cos (x) = \frac{61 - 1}{10}

cos (x) = - \frac{1 + 61}{10} : x = arccos (- \frac{1 + 61}{10}) + 2 πn, x = - arccos (- \frac{1 + 61}{10}) + 2 πn

cos (x) = - \frac{1 + 61}{10}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1 + 61}{10}

以下の一般解

cos (x) = - \frac{1 + 61}{10}

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

x = arccos (- \frac{1 + 61}{10}) + 2 πn, x = - arccos (- \frac{1 + 61}{10}) + 2 πn

x = arccos (- \frac{1 + 61}{10}) + 2 πn, x = - arccos (- \frac{1 + 61}{10}) + 2 πn

cos (x) = \frac{61 - 1}{10} : x = arccos (\frac{61 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{61 - 1}{10}) + 2 πn

cos (x) = \frac{61 - 1}{10}

三角関数の逆数プロパティを適用する

cos (x) = \frac{61 - 1}{10}

以下の一般解

cos (x) = \frac{61 - 1}{10}

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

x = arccos (\frac{61 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{61 - 1}{10}) + 2 πn

x = arccos (\frac{61 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{61 - 1}{10}) + 2 πn

すべての解を組み合わせる

x = arccos (- \frac{1 + 61}{10}) + 2 πn, x = - arccos (- \frac{1 + 61}{10}) + 2 πn, x = arccos (\frac{61 - 1}{10}) + 2 πn, x = 2 π - arccos (\frac{61 - 1}{10}) + 2 πn

10進法形式で解を証明する

x = 2.64882 \dots + 2 πn, x = - 2.64882 \dots + 2 πn, x = 0.82163 \dots + 2 πn, x = 2 π - 0.82163 \dots + 2 πn

グラフ

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