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人気のある三角関数 >

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

解

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

解

x = 0.80515 \dots + 2 πn, x = 2 π - 0.80515 \dots + 2 πn

+1

度

x = 46.13190 \dots^{\circ} + 36 0^{\circ} n, x = 313.86809 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

三角関数の公式を使用して書き換える

- 2 + 2 cos (2 x) + 3 cos (x)

2倍角の公式を使用:

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

= - 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x)

簡素化

- 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x) : 4 cos^{2} (x) + 3 cos (x) - 4

- 2 + 2 (2 cos^{2} (x) - 1) + 3 cos (x)

拡張

2 (2 cos^{2} (x) - 1) : 4 cos^{2} (x) - 2

2 (2 cos^{2} (x) - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = 2 cos^{2} (x), c = 1

= 2 \cdot 2 cos^{2} (x) - 2 \cdot 1

簡素化

2 \cdot 2 cos^{2} (x) - 2 \cdot 1 : 4 cos^{2} (x) - 2

2 \cdot 2 cos^{2} (x) - 2 \cdot 1

数を乗じる:

2 \cdot 2 = 4

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

数を乗じる:

2 \cdot 1 = 2

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

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

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

簡素化

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

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

条件のようなグループ

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

数を引く:

- 2 - 2 = - 4

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

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

- 4 + 3 u + 4 u^{2} = 0 : u = \frac{- 3 + 73}{8}, u = \frac{- 3 - 73}{8}

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

標準的な形式で書く

a x^{2} + b x + c = 0

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

3^{2} - 4 \cdot 4 (- 4) = 73

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

規則を適用

- (- a) = a

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

数を乗じる:

4 \cdot 4 \cdot 4 = 64

= 3^{2} + 64

3^{2} = 9

= 9 + 64

数を足す:

9 + 64 = 73

= 73

u_{1, 2} = \frac{- 3 \pm 73}{2 \cdot 4}

解を分離する

u_{1} = \frac{- 3 + 73}{2 \cdot 4}, u_{2} = \frac{- 3 - 73}{2 \cdot 4}

u = \frac{- 3 + 73}{2 \cdot 4} : \frac{- 3 + 73}{8}

\frac{- 3 + 73}{2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{- 3 + 73}{8}

u = \frac{- 3 - 73}{2 \cdot 4} : \frac{- 3 - 73}{8}

\frac{- 3 - 73}{2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{- 3 - 73}{8}

二次equationの解:

u = \frac{- 3 + 73}{8}, u = \frac{- 3 - 73}{8}

代用を戻す

u = cos (x)

cos (x) = \frac{- 3 + 73}{8}, cos (x) = \frac{- 3 - 73}{8}

cos (x) = \frac{- 3 + 73}{8}, cos (x) = \frac{- 3 - 73}{8}

cos (x) = \frac{- 3 + 73}{8} : x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

cos (x) = \frac{- 3 + 73}{8}

三角関数の逆数プロパティを適用する

cos (x) = \frac{- 3 + 73}{8}

以下の一般解

cos (x) = \frac{- 3 + 73}{8}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

cos (x) = \frac{- 3 - 73}{8} :

解なし

cos (x) = \frac{- 3 - 73}{8}

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

x = arccos (\frac{- 3 + 73}{8}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 73}{8}) + 2 πn

10進法形式で解を証明する

x = 0.80515 \dots + 2 πn, x = 2 π - 0.80515 \dots + 2 πn

グラフ

人気の例

sqrt(6)=-sqrt(8)cos(3x)

6 = - 8 cos (3 x)

sin(θ)=0.12,8<= θ<360

sin (θ) = 0.12, 8 \leq θ < 36 0^{\circ}

2cos(7x)-sqrt(3)=0

2 cos (7 x) - 3 = 0

sec(x)-1=0,0<= x<= 2pi

sec (x) - 1 = 0, 0 \leq x \leq 2 π

(sin(95)}{8.95}=\frac{sin(x))/5

\frac{sin ( 9 5 ^{\circ} )}{8.95} = \frac{sin ( x )}{5}