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人気のある三角関数 >

cos(x)(2cos(x)+1)=3

解

cos (x) (2 cos (x) + 1) = 3

解

x = 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n

解答ステップ

cos (x) (2 cos (x) + 1) = 3

置換で解く

cos (x) (2 cos (x) + 1) = 3

仮定:

cos (x) = u

u (2 u + 1) = 3

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

u (2 u + 1) = 3

拡張

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

u (2 u + 1)

分配法則を適用する:

a (b + c) = ab + a c

a = u, b = 2 u, c = 1

= u \cdot 2 u + u \cdot 1

= 2 uu + 1 \cdot u

簡素化

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

2 uu + 1 \cdot u

2 uu = 2 u^{2}

2 uu

指数の規則を適用する:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

数を足す:

1 + 1 = 2

= 2 u^{2}

1 \cdot u = u

1 \cdot u

乗算:

1 \cdot u = u

= u

= 2 u^{2} + u

= 2 u^{2} + u

2 u^{2} + u = 3

3

を左側に移動します

2 u^{2} + u = 3

両辺から

3

を引く

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

簡素化

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 2, b = 1, c = - 3

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

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

1^{2} - 4 \cdot 2 (- 3) = 5

1^{2} - 4 \cdot 2 (- 3)

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 3)

規則を適用

- (- a) = a

= 1 + 4 \cdot 2 \cdot 3

数を乗じる:

4 \cdot 2 \cdot 3 = 24

= 1 + 24

数を足す:

1 + 24 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 \cdot 2}

解を分離する

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

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

\frac{- 1 + 5}{2 \cdot 2}

数を足す/引く:

- 1 + 5 = 4

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

数を乗じる:

2 \cdot 2 = 4

= \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= 1

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

\frac{- 1 - 5}{2 \cdot 2}

数を引く:

- 1 - 5 = - 6

= \frac{- 6}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{- 6}{4}

分数の規則を適用する:

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

= - \frac{6}{4}

共通因数を約分する:

2

= - \frac{3}{2}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = 1 : x = 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 = 0 + 2 πn

x = 0 + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 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 = 2 πn

グラフ

人気の例

sqrt(3)+cot(x)=0

3 + cot (x) = 0

sin (θ) = - \frac{5}{8}

(sin(θ))/(cos(θ))=1

\frac{sin ( θ )}{cos ( θ )} = 1

tan(θ+26.565)=(12.5^2)/(9.81(60))

tan (θ + 26.56 5^{\circ}) = \frac{12. 5 ^{2}}{9.81 ( 60 )}

21sin(t)cos(t)=-6cos(t)

21 sin (t) cos (t) = - 6 cos (t)