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人気のある三角関数 >

6sin^2(x)+9sin(x)+3=0

解

6 sin^{2} (x) + 9 sin (x) + 3 = 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

6 sin^{2} (x) + 9 sin (x) + 3 = 0

置換で解く

6 sin^{2} (x) + 9 sin (x) + 3 = 0

仮定:

sin (x) = u

6 u^{2} + 9 u + 3 = 0

6 u^{2} + 9 u + 3 = 0 : u = - \frac{1}{2}, u = - 1

6 u^{2} + 9 u + 3 = 0

解くとthe二次式

6 u^{2} + 9 u + 3 = 0

二次Equationの公式:

次の場合:

a = 6, b = 9, c = 3

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 6 \cdot 3}{2 \cdot 6}

9^{2} - 4 \cdot 6 \cdot 3 = 3

9^{2} - 4 \cdot 6 \cdot 3

数を乗じる:

4 \cdot 6 \cdot 3 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

数を引く:

81 - 72 = 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{- 9 \pm 3}{2 \cdot 6}

解を分離する

u_{1} = \frac{- 9 + 3}{2 \cdot 6}, u_{2} = \frac{- 9 - 3}{2 \cdot 6}

u = \frac{- 9 + 3}{2 \cdot 6} : - \frac{1}{2}

\frac{- 9 + 3}{2 \cdot 6}

数を足す/引く:

- 9 + 3 = - 6

= \frac{- 6}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{- 6}{12}

分数の規則を適用する:

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

= - \frac{6}{12}

共通因数を約分する:

6

= - \frac{1}{2}

u = \frac{- 9 - 3}{2 \cdot 6} : - 1

\frac{- 9 - 3}{2 \cdot 6}

数を引く:

- 9 - 3 = - 12

= \frac{- 12}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{- 12}{12}

分数の規則を適用する:

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

= - \frac{12}{12}

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

人気の例

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

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

tan (θ) = 0.1

1 - 2 sin^{2} (x) = 0

8 sin^{2} (θ) = 6

72sin^2(x)=72-36cos(x)

72 sin^{2} (x) = 72 - 36 cos (x)