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人気のある三角関数 >

2sin(2x)sin(x)-3cos(x)=0

解

2 sin (2 x) sin (x) - 3 cos (x) = 0

解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 sin (2 x) sin (x) - 3 cos (x) = 0

三角関数の公式を使用して書き換える

- 3 cos (x) + 2 sin (2 x) sin (x)

2倍角の公式を使用:

sin (2 x) = 2 sin (x) cos (x)

= - 3 cos (x) + 2 \cdot 2 sin (x) cos (x) sin (x)

2 \cdot 2 sin (x) cos (x) sin (x) = 4 sin^{2} (x) cos (x)

2 \cdot 2 sin (x) cos (x) sin (x)

数を乗じる:

2 \cdot 2 = 4

= 4 sin (x) cos (x) sin (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 4 cos (x) sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 4 cos (x) sin^{2} (x)

= - 3 cos (x) + 4 sin^{2} (x) cos (x)

- 3 cos (x) + 4 cos (x) sin^{2} (x) = 0

因数

- 3 cos (x) + 4 cos (x) sin^{2} (x) : cos (x) (2 sin (x) + 3) (2 sin (x) - 3)

- 3 cos (x) + 4 cos (x) sin^{2} (x)

共通項をくくり出す

cos (x)

= cos (x) (- 3 + 4 sin^{2} (x))

因数

4 sin^{2} (x) - 3 : (2 sin (x) + 3) (2 sin (x) - 3)

4 sin^{2} (x) - 3

4 sin^{2} (x) - 3

を書き換え

(2 sin (x))^{2} - (3)^{2}

4 sin^{2} (x) - 3

4

を書き換え

2^{2}

= 2^{2} sin^{2} (x) - 3

累乗根の規則を適用する:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} sin^{2} (x) - (3)^{2}

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

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

= (2 sin (x))^{2} - (3)^{2}

= (2 sin (x))^{2} - (3)^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(2 sin (x))^{2} - (3)^{2} = (2 sin (x) + 3) (2 sin (x) - 3)

= (2 sin (x) + 3) (2 sin (x) - 3)

= cos (x) (2 sin (x) + 3) (2 sin (x) - 3)

cos (x) (2 sin (x) + 3) (2 sin (x) - 3) = 0

各部分を別個に解く

cos (x) = 0 or 2 sin (x) + 3 = 0 or 2 sin (x) - 3 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

以下の一般解

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

2 sin (x) + 3 = 0 : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 sin (x) + 3 = 0

3

を右側に移動します

2 sin (x) + 3 = 0

両辺から

3

を引く

2 sin (x) + 3 - 3 = 0 - 3

簡素化

2 sin (x) = - 3

2 sin (x) = - 3

以下で両辺を割る

2

2 sin (x) = - 3

以下で両辺を割る

2

\frac{2 sin ( x )}{2} = \frac{- 3}{2}

簡素化

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

以下の一般解

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 sin (x) - 3 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

2 sin (x) - 3 = 0

3

を右側に移動します

2 sin (x) - 3 = 0

両辺に

3

を足す

2 sin (x) - 3 + 3 = 0 + 3

簡素化

2 sin (x) = 3

2 sin (x) = 3

以下で両辺を割る

2

2 sin (x) = 3

以下で両辺を割る

2

\frac{2 sin ( x )}{2} = \frac{3}{2}

簡素化

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2}

以下の一般解

sin (x) = \frac{3}{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}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

すべての解を組み合わせる

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

グラフ

人気の例

sin(θ)=(sqrt(13))/7

sin (θ) = \frac{13}{7}

sin^2(θ)-cos^2(θ)=1

sin^{2} (θ) - cos^{2} (θ) = 1

2sin(3θ)=sqrt(2)

2 sin (3 θ) = 2

cot^2(x)+2cot(x)+1=0

cot^{2} (x) + 2 cot (x) + 1 = 0

(cot(x)+1)(sqrt(3)cot(x)+1)=0

(cot (x) + 1) (3 cot (x) + 1) = 0