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人気のある三角関数 >

5cos^2(x)+3cos(x)-1=0

解

5 cos^{2} (x) + 3 cos (x) - 1 = 0

解

x = 1.32995 \dots + 2 πn, x = 2 π - 1.32995 \dots + 2 πn, x = 2.56535 \dots + 2 πn, x = - 2.56535 \dots + 2 πn

+1

度

x = 76.20100 \dots^{\circ} + 36 0^{\circ} n, x = 283.79899 \dots^{\circ} + 36 0^{\circ} n, x = 146.98379 \dots^{\circ} + 36 0^{\circ} n, x = - 146.98379 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

5 cos^{2} (x) + 3 cos (x) - 1 = 0

置換で解く

5 cos^{2} (x) + 3 cos (x) - 1 = 0

仮定:

cos (x) = u

5 u^{2} + 3 u - 1 = 0

5 u^{2} + 3 u - 1 = 0 : u = \frac{- 3 + 29}{10}, u = \frac{- 3 - 29}{10}

5 u^{2} + 3 u - 1 = 0

解くとthe二次式

5 u^{2} + 3 u - 1 = 0

二次Equationの公式:

次の場合:

a = 5, b = 3, c = - 1

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 5 ( - 1 )}{2 \cdot 5}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 5 ( - 1 )}{2 \cdot 5}

3^{2} - 4 \cdot 5 (- 1) = 29

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

規則を適用

- (- a) = a

= 3^{2} + 4 \cdot 5 \cdot 1

数を乗じる:

4 \cdot 5 \cdot 1 = 20

= 3^{2} + 20

3^{2} = 9

= 9 + 20

数を足す:

9 + 20 = 29

= 29

u_{1, 2} = \frac{- 3 \pm 29}{2 \cdot 5}

解を分離する

u_{1} = \frac{- 3 + 29}{2 \cdot 5}, u_{2} = \frac{- 3 - 29}{2 \cdot 5}

u = \frac{- 3 + 29}{2 \cdot 5} : \frac{- 3 + 29}{10}

\frac{- 3 + 29}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 3 + 29}{10}

u = \frac{- 3 - 29}{2 \cdot 5} : \frac{- 3 - 29}{10}

\frac{- 3 - 29}{2 \cdot 5}

数を乗じる:

2 \cdot 5 = 10

= \frac{- 3 - 29}{10}

二次equationの解:

u = \frac{- 3 + 29}{10}, u = \frac{- 3 - 29}{10}

代用を戻す

u = cos (x)

cos (x) = \frac{- 3 + 29}{10}, cos (x) = \frac{- 3 - 29}{10}

cos (x) = \frac{- 3 + 29}{10}, cos (x) = \frac{- 3 - 29}{10}

cos (x) = \frac{- 3 + 29}{10} : x = arccos (\frac{- 3 + 29}{10}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 29}{10}) + 2 πn

cos (x) = \frac{- 3 + 29}{10}

三角関数の逆数プロパティを適用する

cos (x) = \frac{- 3 + 29}{10}

以下の一般解

cos (x) = \frac{- 3 + 29}{10}

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

x = arccos (\frac{- 3 + 29}{10}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 29}{10}) + 2 πn

x = arccos (\frac{- 3 + 29}{10}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 29}{10}) + 2 πn

cos (x) = \frac{- 3 - 29}{10} : x = arccos (\frac{- 3 - 29}{10}) + 2 πn, x = - arccos (\frac{- 3 - 29}{10}) + 2 πn

cos (x) = \frac{- 3 - 29}{10}

三角関数の逆数プロパティを適用する

cos (x) = \frac{- 3 - 29}{10}

以下の一般解

cos (x) = \frac{- 3 - 29}{10}

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

x = arccos (\frac{- 3 - 29}{10}) + 2 πn, x = - arccos (\frac{- 3 - 29}{10}) + 2 πn

x = arccos (\frac{- 3 - 29}{10}) + 2 πn, x = - arccos (\frac{- 3 - 29}{10}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{- 3 + 29}{10}) + 2 πn, x = 2 π - arccos (\frac{- 3 + 29}{10}) + 2 πn, x = arccos (\frac{- 3 - 29}{10}) + 2 πn, x = - arccos (\frac{- 3 - 29}{10}) + 2 πn

10進法形式で解を証明する

x = 1.32995 \dots + 2 πn, x = 2 π - 1.32995 \dots + 2 πn, x = 2.56535 \dots + 2 πn, x = - 2.56535 \dots + 2 πn

グラフ

人気の例

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