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人気のある三角関数 >

8sin^2(x)-10cos(x)-11=0

解

8 sin^{2} (x) - 10 cos (x) - 11 = 0

解

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度

x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

解答ステップ

8 sin^{2} (x) - 10 cos (x) - 11 = 0

三角関数の公式を使用して書き換える

- 11 - 10 cos (x) + 8 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)

= - 11 - 10 cos (x) + 8 (1 - cos^{2} (x))

簡素化

- 11 - 10 cos (x) + 8 (1 - cos^{2} (x)) : - 8 cos^{2} (x) - 10 cos (x) - 3

- 11 - 10 cos (x) + 8 (1 - cos^{2} (x))

拡張

8 (1 - cos^{2} (x)) : 8 - 8 cos^{2} (x)

8 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 8, b = 1, c = cos^{2} (x)

= 8 \cdot 1 - 8 cos^{2} (x)

数を乗じる:

8 \cdot 1 = 8

= 8 - 8 cos^{2} (x)

= - 11 - 10 cos (x) + 8 - 8 cos^{2} (x)

簡素化

- 11 - 10 cos (x) + 8 - 8 cos^{2} (x) : - 8 cos^{2} (x) - 10 cos (x) - 3

- 11 - 10 cos (x) + 8 - 8 cos^{2} (x)

条件のようなグループ

= - 10 cos (x) - 8 cos^{2} (x) - 11 + 8

数を足す/引く:

- 11 + 8 = - 3

= - 8 cos^{2} (x) - 10 cos (x) - 3

= - 8 cos^{2} (x) - 10 cos (x) - 3

= - 8 cos^{2} (x) - 10 cos (x) - 3

- 3 - 10 cos (x) - 8 cos^{2} (x) = 0

置換で解く

- 3 - 10 cos (x) - 8 cos^{2} (x) = 0

仮定:

cos (x) = u

- 3 - 10 u - 8 u^{2} = 0

- 3 - 10 u - 8 u^{2} = 0 : u = - \frac{3}{4}, u = - \frac{1}{2}

- 3 - 10 u - 8 u^{2} = 0

標準的な形式で書く

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

- 8 u^{2} - 10 u - 3 = 0

解くとthe二次式

- 8 u^{2} - 10 u - 3 = 0

二次Equationの公式:

次の場合:

a = - 8, b = - 10, c = - 3

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 ( - 8 ) ( - 3 )}{2 ( - 8 )}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 ( - 8 ) ( - 3 )}{2 ( - 8 )}

(- 10)^{2} - 4 (- 8) (- 3) = 2

(- 10)^{2} - 4 (- 8) (- 3)

規則を適用

- (- a) = a

= (- 10)^{2} - 4 \cdot 8 \cdot 3

指数の規則を適用する:

n

が偶数であれば

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

(- 10)^{2} = 1 0^{2}

= 1 0^{2} - 4 \cdot 8 \cdot 3

数を乗じる:

4 \cdot 8 \cdot 3 = 96

= 1 0^{2} - 96

1 0^{2} = 100

= 100 - 96

数を引く:

100 - 96 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 10 ) \pm 2}{2 ( - 8 )}

解を分離する

u_{1} = \frac{- ( - 10 ) + 2}{2 ( - 8 )}, u_{2} = \frac{- ( - 10 ) - 2}{2 ( - 8 )}

u = \frac{- ( - 10 ) + 2}{2 ( - 8 )} : - \frac{3}{4}

\frac{- ( - 10 ) + 2}{2 ( - 8 )}

括弧を削除する:

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

= \frac{10 + 2}{- 2 \cdot 8}

数を足す:

10 + 2 = 12

= \frac{12}{- 2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{12}{- 16}

分数の規則を適用する:

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

= - \frac{12}{16}

共通因数を約分する:

4

= - \frac{3}{4}

u = \frac{- ( - 10 ) - 2}{2 ( - 8 )} : - \frac{1}{2}

\frac{- ( - 10 ) - 2}{2 ( - 8 )}

括弧を削除する:

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

= \frac{10 - 2}{- 2 \cdot 8}

数を引く:

10 - 2 = 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{- 16}

分数の規則を適用する:

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

= - \frac{8}{16}

共通因数を約分する:

8

= - \frac{1}{2}

二次equationの解:

u = - \frac{3}{4}, u = - \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = - \frac{3}{4}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{3}{4}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{3}{4} : x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

cos (x) = - \frac{3}{4}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{3}{4}

以下の一般解

cos (x) = - \frac{3}{4}

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

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

以下の一般解

cos (x) = - \frac{1}{2}

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

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

すべての解を組み合わせる

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

10進法形式で解を証明する

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

グラフ

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