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人気のある三角関数 >

cos^4(x)= 1/16

解

cos^{4} (x) = \frac{1}{16}

解

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

+1

度

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

解答ステップ

cos^{4} (x) = \frac{1}{16}

置換で解く

cos^{4} (x) = \frac{1}{16}

仮定:

cos (x) = u

u^{4} = \frac{1}{16}

u^{4} = \frac{1}{16} : u = \frac{1}{2}, u = - \frac{1}{2}, u = i \frac{1}{2}, u = - i \frac{1}{2}

u^{4} = \frac{1}{16}

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

v^{2} = \frac{1}{16}

解く

v^{2} = \frac{1}{16} : v = \frac{1}{16}, v = - \frac{1}{16}

v^{2} = \frac{1}{16}

(g (x))^{2} = f (a)

の場合, 解は

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

v = \frac{1}{16}, v = - \frac{1}{16}

v = \frac{1}{16}, v = - \frac{1}{16}

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = \frac{1}{16} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{16}

簡素化

\frac{1}{16} : \frac{1}{4}

\frac{1}{16}

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}{16}

16 = 4

16

数を因数に分解する:

16 = 4^{2}

= 4^{2}

累乗根の規則を適用する:

n a^{n} = a

4^{2} = 4

= 4

= \frac{1}{4}

規則を適用

1 = 1

= \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}

解く

u^{2} = - \frac{1}{16} : u = i \frac{1}{2}, u = - i \frac{1}{2}

u^{2} = - \frac{1}{16}

簡素化

- \frac{1}{16} : - \frac{1}{4}

- \frac{1}{16}

簡素化

\frac{1}{16} : \frac{1}{4}

\frac{1}{16}

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}{16}

16 = 4

16

数を因数に分解する:

16 = 4^{2}

= 4^{2}

累乗根の規則を適用する:

n a^{n} = a

4^{2} = 4

= 4

= \frac{1}{4}

= - \frac{1}{4}

規則を適用

1 = 1

= - \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}

簡素化

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

- \frac{1}{4}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{4}

虚数の規則を適用する:

- 1 = i

= i \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} = \frac{1}{4}

= i \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

= i \frac{1}{2}

規則を適用

1 = 1

= i \frac{1}{2}

標準的な複素数形式で

i \frac{1}{2}

を書き換える:

\frac{1}{2} i

i \frac{1}{2}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 i}{2}

乗算:

1 i = i

= \frac{i}{2}

= \frac{1}{2} i

簡素化

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

- - \frac{1}{4}

簡素化

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

- \frac{1}{4}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{4}

虚数の規則を適用する:

- 1 = i

= i \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} = \frac{1}{4}

= i \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

= i \frac{1}{2}

= - i \frac{1}{2}

規則を適用

1 = 1

= - \frac{1}{2} i

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

解答は

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

代用を戻す

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

以下の一般解

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

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}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

以下の一般解

cos (x) = - \frac{1}{2}

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

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

cos (x) = i \frac{1}{2} :

解なし

cos (x) = i \frac{1}{2}

解なし

cos (x) = - i \frac{1}{2} :

解なし

cos (x) = - i \frac{1}{2}

解なし

すべての解を組み合わせる

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

グラフ

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