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人気のある三角関数 >

4(cos(x)+1)cos(x)=3

解

4 (cos (x) + 1) cos (x) = 3

解

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

+1

度

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

解答ステップ

4 (cos (x) + 1) cos (x) = 3

置換で解く

4 (cos (x) + 1) cos (x) = 3

仮定:

cos (x) = u

4 (u + 1) u = 3

4 (u + 1) u = 3 : u = \frac{1}{2}, u = - \frac{3}{2}

4 (u + 1) u = 3

拡張

4 (u + 1) u : 4 u^{2} + 4 u

4 (u + 1) u

= 4 u (u + 1)

分配法則を適用する:

a (b + c) = ab + a c

a = 4 u, b = u, c = 1

= 4 uu + 4 u \cdot 1

= 4 uu + 4 \cdot 1 \cdot u

簡素化

4 uu + 4 \cdot 1 \cdot u : 4 u^{2} + 4 u

4 uu + 4 \cdot 1 \cdot u

4 uu = 4 u^{2}

4 uu

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 4 u^{1 + 1}

数を足す:

1 + 1 = 2

= 4 u^{2}

4 \cdot 1 \cdot u = 4 u

4 \cdot 1 \cdot u

数を乗じる:

4 \cdot 1 = 4

= 4 u

= 4 u^{2} + 4 u

= 4 u^{2} + 4 u

4 u^{2} + 4 u = 3

3

を左側に移動します

4 u^{2} + 4 u = 3

両辺から

3

を引く

4 u^{2} + 4 u - 3 = 3 - 3

簡素化

4 u^{2} + 4 u - 3 = 0

4 u^{2} + 4 u - 3 = 0

解くとthe二次式

4 u^{2} + 4 u - 3 = 0

二次Equationの公式:

次の場合:

a = 4, b = 4, c = - 3

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

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

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

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

規則を適用

- (- a) = a

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

数を乗じる:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数を足す:

16 + 48 = 64

= 64

数を因数に分解する:

64 = 8^{2}

= 8^{2}

累乗根の規則を適用する:

n a^{n} = a

8^{2} = 8

= 8

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

解を分離する

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

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

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

数を足す/引く:

- 4 + 8 = 4

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

数を乗じる:

2 \cdot 4 = 8

= \frac{4}{8}

共通因数を約分する:

4

= \frac{1}{2}

u = \frac{- 4 - 8}{2 \cdot 4} : - \frac{3}{2}

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

数を引く:

- 4 - 8 = - 12

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

数を乗じる:

2 \cdot 4 = 8

= \frac{- 12}{8}

分数の規則を適用する:

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

= - \frac{12}{8}

共通因数を約分する:

4

= - \frac{3}{2}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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) = - \frac{3}{2} :

解なし

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

- 1 \leq cos (x) \leq 1

解なし

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

グラフ

人気の例

sin^2(x)+1=cos^2(x)-2sin^4(x)

sin^{2} (x) + 1 = cos^{2} (x) - 2 sin^{4} (x)

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

cos(2a)= 2/3 ,sin(a)

cos (2 a) = \frac{2}{3}, sin (a)

cot (2 x - 3) - 1 = 0

4sin^{32}(x)-3cos^2(x)=0

4 sin^{32} (x) - 3 cos^{2} (x) = 0