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人気のある三角関数 >

18sin^2(x)+27sin(x)+9=0

解

18 sin^{2} (x) + 27 sin (x) + 9 = 0

解

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

18 sin^{2} (x) + 27 sin (x) + 9 = 0

置換で解く

18 sin^{2} (x) + 27 sin (x) + 9 = 0

仮定:

sin (x) = u

18 u^{2} + 27 u + 9 = 0

18 u^{2} + 27 u + 9 = 0 : u = - \frac{1}{2}, u = - 1

18 u^{2} + 27 u + 9 = 0

解くとthe二次式

18 u^{2} + 27 u + 9 = 0

二次Equationの公式:

次の場合:

a = 18, b = 27, c = 9

u_{1, 2} = \frac{- 27 \pm 2 7 ^{2} - 4 \cdot 18 \cdot 9}{2 \cdot 18}

u_{1, 2} = \frac{- 27 \pm 2 7 ^{2} - 4 \cdot 18 \cdot 9}{2 \cdot 18}

2 7^{2} - 4 \cdot 18 \cdot 9 = 9

2 7^{2} - 4 \cdot 18 \cdot 9

数を乗じる:

4 \cdot 18 \cdot 9 = 648

= 2 7^{2} - 648

2 7^{2} = 729

= 729 - 648

数を引く:

729 - 648 = 81

= 81

数を因数に分解する:

81 = 9^{2}

= 9^{2}

累乗根の規則を適用する:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 27 \pm 9}{2 \cdot 18}

解を分離する

u_{1} = \frac{- 27 + 9}{2 \cdot 18}, u_{2} = \frac{- 27 - 9}{2 \cdot 18}

u = \frac{- 27 + 9}{2 \cdot 18} : - \frac{1}{2}

\frac{- 27 + 9}{2 \cdot 18}

数を足す/引く:

- 27 + 9 = - 18

= \frac{- 18}{2 \cdot 18}

数を乗じる:

2 \cdot 18 = 36

= \frac{- 18}{36}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{18}{36}

共通因数を約分する:

18

= - \frac{1}{2}

u = \frac{- 27 - 9}{2 \cdot 18} : - 1

\frac{- 27 - 9}{2 \cdot 18}

数を引く:

- 27 - 9 = - 36

= \frac{- 36}{2 \cdot 18}

数を乗じる:

2 \cdot 18 = 36

= \frac{- 36}{36}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{36}{36}

規則を適用

\frac{a}{a} = 1

= - 1

二次equationの解:

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

代用を戻す

u = sin (x)

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

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

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

以下の一般解

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

すべての解を組み合わせる

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

グラフ

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