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人気のある三角関数 >

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

解

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

解

x = 1.11692 \dots + 2 πn, x = 2 π - 1.11692 \dots + 2 πn

+1

度

x = 63.99515 \dots^{\circ} + 36 0^{\circ} n, x = 296.00484 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

cos (x) = u

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

u^{2} - 5 u + 2 = 0 : u = \frac{5 + 17}{2}, u = \frac{5 - 17}{2}

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 1, b = - 5, c = 2

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

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

(- 5)^{2} - 4 \cdot 1 \cdot 2 = 17

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

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 2

数を乗じる:

4 \cdot 1 \cdot 2 = 8

= 5^{2} - 8

5^{2} = 25

= 25 - 8

数を引く:

25 - 8 = 17

= 17

u_{1, 2} = \frac{- ( - 5 ) \pm 17}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 5 ) + 17}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 17}{2 \cdot 1}

u = \frac{- ( - 5 ) + 17}{2 \cdot 1} : \frac{5 + 17}{2}

\frac{- ( - 5 ) + 17}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 + 17}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{5 + 17}{2}

u = \frac{- ( - 5 ) - 17}{2 \cdot 1} : \frac{5 - 17}{2}

\frac{- ( - 5 ) - 17}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 - 17}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{5 - 17}{2}

二次equationの解:

u = \frac{5 + 17}{2}, u = \frac{5 - 17}{2}

代用を戻す

u = cos (x)

cos (x) = \frac{5 + 17}{2}, cos (x) = \frac{5 - 17}{2}

cos (x) = \frac{5 + 17}{2}, cos (x) = \frac{5 - 17}{2}

cos (x) = \frac{5 + 17}{2} :

解なし

cos (x) = \frac{5 + 17}{2}

- 1 \leq cos (x) \leq 1

解なし

cos (x) = \frac{5 - 17}{2} : x = arccos (\frac{5 - 17}{2}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{2}) + 2 πn

cos (x) = \frac{5 - 17}{2}

三角関数の逆数プロパティを適用する

cos (x) = \frac{5 - 17}{2}

以下の一般解

cos (x) = \frac{5 - 17}{2}

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

x = arccos (\frac{5 - 17}{2}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{2}) + 2 πn

x = arccos (\frac{5 - 17}{2}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{2}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{5 - 17}{2}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{2}) + 2 πn

10進法形式で解を証明する

x = 1.11692 \dots + 2 πn, x = 2 π - 1.11692 \dots + 2 πn

グラフ

人気の例

2sin^2(t)cos(t)=0

2 sin^{2} (t) cos (t) = 0

5sin(x)-12cos(x)=13

5 sin (x) - 12 cos (x) = 13

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2 cos (\frac{3 x - π}{6}) + 1 = 0, 0 < x < 2 π

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