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人気のある三角関数 >

2cos^2(x)-15cos(x)+7=0

解

2 cos^{2} (x) - 15 cos (x) + 7 = 0

解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 cos^{2} (x) - 15 cos (x) + 7 = 0

置換で解く

2 cos^{2} (x) - 15 cos (x) + 7 = 0

仮定:

cos (x) = u

2 u^{2} - 15 u + 7 = 0

2 u^{2} - 15 u + 7 = 0 : u = 7, u = \frac{1}{2}

2 u^{2} - 15 u + 7 = 0

解くとthe二次式

2 u^{2} - 15 u + 7 = 0

二次Equationの公式:

次の場合:

a = 2, b = - 15, c = 7

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 \cdot 2 \cdot 7}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 \cdot 2 \cdot 7}{2 \cdot 2}

(- 15)^{2} - 4 \cdot 2 \cdot 7 = 13

(- 15)^{2} - 4 \cdot 2 \cdot 7

指数の規則を適用する:

n

が偶数であれば

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

(- 15)^{2} = 1 5^{2}

= 1 5^{2} - 4 \cdot 2 \cdot 7

数を乗じる:

4 \cdot 2 \cdot 7 = 56

= 1 5^{2} - 56

1 5^{2} = 225

= 225 - 56

数を引く:

225 - 56 = 169

= 169

数を因数に分解する:

169 = 1 3^{2}

= 1 3^{2}

累乗根の規則を適用する:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 15 ) \pm 13}{2 \cdot 2}

解を分離する

u_{1} = \frac{- ( - 15 ) + 13}{2 \cdot 2}, u_{2} = \frac{- ( - 15 ) - 13}{2 \cdot 2}

u = \frac{- ( - 15 ) + 13}{2 \cdot 2} : 7

\frac{- ( - 15 ) + 13}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{15 + 13}{2 \cdot 2}

数を足す:

15 + 13 = 28

= \frac{28}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{28}{4}

数を割る:

\frac{28}{4} = 7

= 7

u = \frac{- ( - 15 ) - 13}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 15 ) - 13}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{15 - 13}{2 \cdot 2}

数を引く:

15 - 13 = 2

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

数を乗じる:

2 \cdot 2 = 4

= \frac{2}{4}

共通因数を約分する:

2

= \frac{1}{2}

二次equationの解:

u = 7, u = \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = 7, cos (x) = \frac{1}{2}

cos (x) = 7, cos (x) = \frac{1}{2}

cos (x) = 7 :

解なし

cos (x) = 7

- 1 \leq cos (x) \leq 1

解なし

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

以下の一般解

cos (x) = \frac{1}{2}

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

すべての解を組み合わせる

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

グラフ

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