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人気のある三角関数 >

8cos^2(x)+4cos(x)-4=0

解

8 cos^{2} (x) + 4 cos (x) - 4 = 0

解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

8 cos^{2} (x) + 4 cos (x) - 4 = 0

置換で解く

8 cos^{2} (x) + 4 cos (x) - 4 = 0

仮定:

cos (x) = u

8 u^{2} + 4 u - 4 = 0

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

8 u^{2} + 4 u - 4 = 0

解くとthe二次式

8 u^{2} + 4 u - 4 = 0

二次Equationの公式:

次の場合:

a = 8, b = 4, c = - 4

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

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

4^{2} - 4 \cdot 8 (- 4) = 12

4^{2} - 4 \cdot 8 (- 4)

規則を適用

- (- a) = a

= 4^{2} + 4 \cdot 8 \cdot 4

数を乗じる:

4 \cdot 8 \cdot 4 = 128

= 4^{2} + 128

4^{2} = 16

= 16 + 128

数を足す:

16 + 128 = 144

= 144

数を因数に分解する:

144 = 1 2^{2}

= 1 2^{2}

累乗根の規則を適用する:

n a^{n} = a

1 2^{2} = 12

= 12

u_{1, 2} = \frac{- 4 \pm 12}{2 \cdot 8}

解を分離する

u_{1} = \frac{- 4 + 12}{2 \cdot 8}, u_{2} = \frac{- 4 - 12}{2 \cdot 8}

u = \frac{- 4 + 12}{2 \cdot 8} : \frac{1}{2}

\frac{- 4 + 12}{2 \cdot 8}

数を足す/引く:

- 4 + 12 = 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{16}

共通因数を約分する:

8

= \frac{1}{2}

u = \frac{- 4 - 12}{2 \cdot 8} : - 1

\frac{- 4 - 12}{2 \cdot 8}

数を引く:

- 4 - 12 = - 16

= \frac{- 16}{2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 16}{16}

分数の規則を適用する:

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

= - \frac{16}{16}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn

グラフ

人気の例

sin^2(x)=cos(x)

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

1 - cos (x) = 0

2sin(x)cos(x)=-3cos(x)

2 sin (x) cos (x) = - 3 cos (x)

sin(x/2)=cos(x/2)

sin (\frac{x}{2}) = cos (\frac{x}{2})

tan^2(x)=1-sec(x)

tan^{2} (x) = 1 - sec (x)