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人気のある三角関数 >

6sin^2(a)+cos(2a)=2

解

6 sin^{2} (a) + cos (2 a) = 2

解

a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn, a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn

+1

度

a = 3 0^{\circ} + 36 0^{\circ} n, a = 15 0^{\circ} + 36 0^{\circ} n, a = 21 0^{\circ} + 36 0^{\circ} n, a = 33 0^{\circ} + 36 0^{\circ} n

解答ステップ

6 sin^{2} (a) + cos (2 a) = 2

両辺から

2

を引く

6 sin^{2} (a) + cos (2 a) - 2 = 0

三角関数の公式を使用して書き換える

- 2 + cos (2 a) + 6 sin^{2} (a)

2倍角の公式を使用:

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

= - 2 + 1 - 2 sin^{2} (a) + 6 sin^{2} (a)

簡素化

- 2 + 1 - 2 sin^{2} (a) + 6 sin^{2} (a) : 4 sin^{2} (a) - 1

- 2 + 1 - 2 sin^{2} (a) + 6 sin^{2} (a)

類似した元を足す:

- 2 sin^{2} (a) + 6 sin^{2} (a) = 4 sin^{2} (a)

= - 2 + 1 + 4 sin^{2} (a)

数を足す/引く:

- 2 + 1 = - 1

= 4 sin^{2} (a) - 1

= 4 sin^{2} (a) - 1

- 1 + 4 sin^{2} (a) = 0

置換で解く

- 1 + 4 sin^{2} (a) = 0

仮定:

sin (a) = u

- 1 + 4 u^{2} = 0

- 1 + 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 1 + 4 u^{2} = 0

1

を右側に移動します

- 1 + 4 u^{2} = 0

両辺に

1

を足す

- 1 + 4 u^{2} + 1 = 0 + 1

簡素化

4 u^{2} = 1

4 u^{2} = 1

以下で両辺を割る

4

4 u^{2} = 1

以下で両辺を割る

4

\frac{4 u ^{2}}{4} = \frac{1}{4}

簡素化

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

規則を適用

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

簡素化

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

規則を適用

1 = 1

= - \frac{1}{2}

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

代用を戻す

u = sin (a)

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

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

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

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

以下の一般解

sin (a) = \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}

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

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

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

sin (a) = - \frac{1}{2}

以下の一般解

sin (a) = - \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}

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

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

すべての解を組み合わせる

a = \frac{π}{6} + 2 πn, a = \frac{5 π}{6} + 2 πn, a = \frac{7 π}{6} + 2 πn, a = \frac{11 π}{6} + 2 πn

グラフ

人気の例

cos^{(2)}(x)+sin(x)=1

cos^{(2)} (x) + sin (x) = 1

sin (2 q) = 0

tan (x) = \frac{218}{75}

cos (x) - 2 + 2 = 1

- 3 + cos (2 x) 2 = 0