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人気のある三角関数 >

cos^4(x)= 1/2

解

cos^{4} (x) = \frac{1}{2}

解

x = 0.57185 \dots + 2 πn, x = 2 π - 0.57185 \dots + 2 πn, x = 2.56973 \dots + 2 πn, x = - 2.56973 \dots + 2 πn

+1

度

x = 32.76509 \dots^{\circ} + 36 0^{\circ} n, x = 327.23490 \dots^{\circ} + 36 0^{\circ} n, x = 147.23490 \dots^{\circ} + 36 0^{\circ} n, x = - 147.23490 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

cos^{4} (x) = \frac{1}{2}

置換で解く

cos^{4} (x) = \frac{1}{2}

仮定:

cos (x) = u

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

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

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

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

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

解く

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

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

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

の場合, 解は

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

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

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

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

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

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

x^{2} = f (a)

の場合, 解は

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

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

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

\frac{1}{2}

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{\frac{1}{2}}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot \frac{1}{2}}

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

\frac{1}{2} \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1}{2 \cdot 2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{1}{4}

= (\frac{1}{2})^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 4 \frac{1}{2}

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

- \frac{1}{2}

簡素化

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

\frac{1}{2}

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{\frac{1}{2}}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot \frac{1}{2}}

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

\frac{1}{2} \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1}{2 \cdot 2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{1}{4}

= (\frac{1}{2})^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 4 \frac{1}{2}

= - 4 \frac{1}{2}

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

解く

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

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

x^{2} = f (a)

の場合, 解は

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

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

簡素化

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

- \frac{1}{2}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{2}

虚数の規則を適用する:

- 1 = i

= i \frac{1}{2}

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

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{\frac{1}{2}}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot \frac{1}{2}}

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

\frac{1}{2} \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1}{2 \cdot 2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{1}{4}

= (\frac{1}{2})^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 4 \frac{1}{2}

= i 4 \frac{1}{2}

簡素化

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

- - \frac{1}{2}

簡素化

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

- \frac{1}{2}

累乗根の規則を適用する:

- a = - 1 a

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

= - 1 \frac{1}{2}

虚数の規則を適用する:

- 1 = i

= i \frac{1}{2}

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

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{\frac{1}{2}}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot \frac{1}{2}}

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

\frac{1}{2} \cdot \frac{1}{2}

分数を乗じる:

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

= \frac{1 \cdot 1}{2 \cdot 2}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{1}{4}

= (\frac{1}{2})^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 4 \frac{1}{2}

= i 4 \frac{1}{2}

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

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

解答は

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

代用を戻す

u = cos (x)

cos (x) = 4 \frac{1}{2}, cos (x) = - 4 \frac{1}{2}, cos (x) = i 4 \frac{1}{2}, cos (x) = - i 4 \frac{1}{2}

cos (x) = 4 \frac{1}{2}, cos (x) = - 4 \frac{1}{2}, cos (x) = i 4 \frac{1}{2}, cos (x) = - i 4 \frac{1}{2}

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

cos (x) = 4 \frac{1}{2}

三角関数の逆数プロパティを適用する

cos (x) = 4 \frac{1}{2}

以下の一般解

cos (x) = 4 \frac{1}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

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

cos (x) = - 4 \frac{1}{2}

三角関数の逆数プロパティを適用する

cos (x) = - 4 \frac{1}{2}

以下の一般解

cos (x) = - 4 \frac{1}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- 4 \frac{1}{2}) + 2 πn, x = - arccos (- 4 \frac{1}{2}) + 2 πn

x = arccos (- 4 \frac{1}{2}) + 2 πn, x = - arccos (- 4 \frac{1}{2}) + 2 πn

cos (x) = i 4 \frac{1}{2} :

解なし

cos (x) = i 4 \frac{1}{2}

解なし

cos (x) = - i 4 \frac{1}{2} :

解なし

cos (x) = - i 4 \frac{1}{2}

解なし

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.57185 \dots + 2 πn, x = 2 π - 0.57185 \dots + 2 πn, x = 2.56973 \dots + 2 πn, x = - 2.56973 \dots + 2 πn

グラフ

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