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人気のある三角関数 >

2sin^2(x)-5sin(x)=-2

解

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

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

sin (x) = u

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

2 u^{2} - 5 u = - 2 : u = 2, u = \frac{1}{2}

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

2

を左側に移動します

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

両辺に

2

を足す

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

簡素化

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

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

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

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数を引く:

25 - 16 = 9

= 9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

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

解を分離する

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

u = \frac{- ( - 5 ) + 3}{2 \cdot 2} : 2

\frac{- ( - 5 ) + 3}{2 \cdot 2}

規則を適用

- (- a) = a

= \frac{5 + 3}{2 \cdot 2}

数を足す:

5 + 3 = 8

= \frac{8}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{8}{4}

数を割る:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 5 ) - 3}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 5 ) - 3}{2 \cdot 2}

規則を適用

- (- a) = a

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

数を引く:

5 - 3 = 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 = 2, u = \frac{1}{2}

代用を戻す

u = sin (x)

sin (x) = 2, sin (x) = \frac{1}{2}

sin (x) = 2, sin (x) = \frac{1}{2}

sin (x) = 2 :

解なし

sin (x) = 2

- 1 \leq sin (x) \leq 1

解なし

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

以下の一般解

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

人気の例

8sin^2(x)+6sin(x)+1=0

8 sin^{2} (x) + 6 sin (x) + 1 = 0

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