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人気のある三角関数 >

10cos^2(x)+9cos(x)+2=0

解

10 cos^{2} (x) + 9 cos (x) + 2 = 0

解

x = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度

x = 113.57817 \dots^{\circ} + 36 0^{\circ} n, x = - 113.57817 \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

10 cos^{2} (x) + 9 cos (x) + 2 = 0

置換で解く

10 cos^{2} (x) + 9 cos (x) + 2 = 0

仮定:

cos (x) = u

10 u^{2} + 9 u + 2 = 0

10 u^{2} + 9 u + 2 = 0 : u = - \frac{2}{5}, u = - \frac{1}{2}

10 u^{2} + 9 u + 2 = 0

解くとthe二次式

10 u^{2} + 9 u + 2 = 0

二次Equationの公式:

次の場合:

a = 10, b = 9, c = 2

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 10 \cdot 2}{2 \cdot 10}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 10 \cdot 2}{2 \cdot 10}

9^{2} - 4 \cdot 10 \cdot 2 = 1

9^{2} - 4 \cdot 10 \cdot 2

数を乗じる:

4 \cdot 10 \cdot 2 = 80

= 9^{2} - 80

9^{2} = 81

= 81 - 80

数を引く:

81 - 80 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- 9 \pm 1}{2 \cdot 10}

解を分離する

u_{1} = \frac{- 9 + 1}{2 \cdot 10}, u_{2} = \frac{- 9 - 1}{2 \cdot 10}

u = \frac{- 9 + 1}{2 \cdot 10} : - \frac{2}{5}

\frac{- 9 + 1}{2 \cdot 10}

数を足す/引く:

- 9 + 1 = - 8

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

数を乗じる:

2 \cdot 10 = 20

= \frac{- 8}{20}

分数の規則を適用する:

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

= - \frac{8}{20}

共通因数を約分する:

4

= - \frac{2}{5}

u = \frac{- 9 - 1}{2 \cdot 10} : - \frac{1}{2}

\frac{- 9 - 1}{2 \cdot 10}

数を引く:

- 9 - 1 = - 10

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

数を乗じる:

2 \cdot 10 = 20

= \frac{- 10}{20}

分数の規則を適用する:

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

= - \frac{10}{20}

共通因数を約分する:

10

= - \frac{1}{2}

二次equationの解:

u = - \frac{2}{5}, u = - \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = - \frac{2}{5}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{2}{5}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{2}{5} : x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

cos (x) = - \frac{2}{5}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{2}{5}

以下の一般解

cos (x) = - \frac{2}{5}

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

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 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{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 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 = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

グラフ

人気の例

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

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

sin (θ) = - 0.4

sin (θ) = - 0.2

6 cos (2 x) = 5

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