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人気のある三角関数 >

cos^2(x)-4cos(x)+3=0

解

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

解

x = 2 πn

+1

度

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

解答ステップ

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

置換で解く

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

仮定:

cos (x) = u

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

u^{2} - 4 u + 3 = 0 : u = 3, u = 1

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 1, b = - 4, c = 3

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

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

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

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

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 4)^{2} = 4^{2}

= 4^{2} - 4 \cdot 1 \cdot 3

数を乗じる:

4 \cdot 1 \cdot 3 = 12

= 4^{2} - 12

4^{2} = 16

= 16 - 12

数を引く:

16 - 12 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

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

解を分離する

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

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

\frac{- ( - 4 ) + 2}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{4 + 2}{2 \cdot 1}

数を足す:

4 + 2 = 6

= \frac{6}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{6}{2}

数を割る:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 4 ) - 2}{2 \cdot 1} : 1

\frac{- ( - 4 ) - 2}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{4 - 2}{2 \cdot 1}

数を引く:

4 - 2 = 2

= \frac{2}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{2}{2}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

u = 3, u = 1

代用を戻す

u = cos (x)

cos (x) = 3, cos (x) = 1

cos (x) = 3, cos (x) = 1

cos (x) = 3 :

解なし

cos (x) = 3

- 1 \leq cos (x) \leq 1

解なし

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

すべての解を組み合わせる

x = 2 πn

グラフ

人気の例

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

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

4 cos^{2} (x) = 2

sin (x) = - \frac{1}{5}

cos(2x)+sin(2x)=0

cos (2 x) + sin (2 x) = 0

sin (x) = \frac{4}{7}