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人気のある三角関数 >

5cos^2(x)+3cos(x)-2=0

解

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

解

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = π + 2 πn

+1

度

x = 66.42182 \dots^{\circ} + 36 0^{\circ} n, x = 293.57817 \dots^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

cos (x) = u

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

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 5, b = 3, c = - 2

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

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

3^{2} - 4 \cdot 5 (- 2) = 7

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

規則を適用

- (- a) = a

= 3^{2} + 4 \cdot 5 \cdot 2

数を乗じる:

4 \cdot 5 \cdot 2 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数を足す:

9 + 40 = 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{- 3 \pm 7}{2 \cdot 5}

解を分離する

u_{1} = \frac{- 3 + 7}{2 \cdot 5}, u_{2} = \frac{- 3 - 7}{2 \cdot 5}

u = \frac{- 3 + 7}{2 \cdot 5} : \frac{2}{5}

\frac{- 3 + 7}{2 \cdot 5}

数を足す/引く:

- 3 + 7 = 4

= \frac{4}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{4}{10}

共通因数を約分する:

2

= \frac{2}{5}

u = \frac{- 3 - 7}{2 \cdot 5} : - 1

\frac{- 3 - 7}{2 \cdot 5}

数を引く:

- 3 - 7 = - 10

= \frac{- 10}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 10}{10}

分数の規則を適用する:

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

= - \frac{10}{10}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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

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

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

以下の一般解

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = π + 2 πn

グラフ

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