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人気のある三角関数 >

81(cos^2(x))=6

解

81 (cos^{2} (x)) = 6

解

x = 1.29515 \dots + 2 πn, x = 2 π - 1.29515 \dots + 2 πn, x = 1.84643 \dots + 2 πn, x = - 1.84643 \dots + 2 πn

+1

度

x = 74.20683 \dots^{\circ} + 36 0^{\circ} n, x = 285.79316 \dots^{\circ} + 36 0^{\circ} n, x = 105.79316 \dots^{\circ} + 36 0^{\circ} n, x = - 105.79316 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

81 (cos^{2} (x)) = 6

置換で解く

81 cos^{2} (x) = 6

仮定:

cos (x) = u

81 u^{2} = 6

81 u^{2} = 6 : u = \frac{6}{9}, u = - \frac{6}{9}

81 u^{2} = 6

以下で両辺を割る

81

81 u^{2} = 6

以下で両辺を割る

81

\frac{81 u ^{2}}{81} = \frac{6}{81}

簡素化

u^{2} = \frac{2}{27}

u^{2} = \frac{2}{27}

x^{2} = f (a)

の場合, 解は

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

u = \frac{2}{27}, u = - \frac{2}{27}

\frac{2}{27} = \frac{6}{9}

\frac{2}{27}

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{2}{27}

27 = 33

27

以下の素因数分解:

27 : 3^{3}

27

27327 = 9 \cdot 3

で割る

= 3 \cdot 9

939 = 3 \cdot 3

で割る

= 3 \cdot 3 \cdot 3

3

は素数なので, さらに因数分解はできない

= 3 \cdot 3 \cdot 3

= 3^{3}

= 3^{3}

指数の規則を適用する:

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

= 3^{2} \cdot 3

累乗根の規則を適用する:

n ab = n a n b

= 3 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 33

= \frac{2}{3 3}

有理化する

\frac{2}{3 3} : \frac{6}{9}

\frac{2}{3 3}

共役で乗じる

\frac{3}{3}

= \frac{2 3}{3 3 3}

23 = 6

23

累乗根の規則を適用する:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数を乗じる:

2 \cdot 3 = 6

= 6

333 = 9

333

指数の規則を適用する:

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

333 = 3 \cdot 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} = 3^{1 + \frac{1}{2} + \frac{1}{2}}

= 3^{1 + \frac{1}{2} + \frac{1}{2}}

類似した元を足す:

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

= 3^{1 + 2 \cdot \frac{1}{2}}

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

2 \cdot \frac{1}{2}

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 3^{1 + 1}

数を足す:

1 + 1 = 2

= 3^{2}

3^{2} = 9

= 9

= \frac{6}{9}

= \frac{6}{9}

- \frac{2}{27} = - \frac{6}{9}

- \frac{2}{27}

簡素化

\frac{2}{27} : \frac{2}{3 3}

\frac{2}{27}

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{2}{27}

27 = 33

27

以下の素因数分解:

27 : 3^{3}

27

27327 = 9 \cdot 3

で割る

= 3 \cdot 9

939 = 3 \cdot 3

で割る

= 3 \cdot 3 \cdot 3

3

は素数なので, さらに因数分解はできない

= 3 \cdot 3 \cdot 3

= 3^{3}

= 3^{3}

指数の規則を適用する:

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

= 3^{2} \cdot 3

累乗根の規則を適用する:

n ab = n a n b

= 3 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 33

= \frac{2}{3 3}

= - \frac{2}{3 3}

有理化する

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

- \frac{2}{3 3}

共役で乗じる

\frac{3}{3}

= - \frac{2 3}{3 3 3}

23 = 6

23

累乗根の規則を適用する:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数を乗じる:

2 \cdot 3 = 6

= 6

333 = 9

333

指数の規則を適用する:

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

333 = 3 \cdot 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} = 3^{1 + \frac{1}{2} + \frac{1}{2}}

= 3^{1 + \frac{1}{2} + \frac{1}{2}}

類似した元を足す:

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

= 3^{1 + 2 \cdot \frac{1}{2}}

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

2 \cdot \frac{1}{2}

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 3^{1 + 1}

数を足す:

1 + 1 = 2

= 3^{2}

3^{2} = 9

= 9

= - \frac{6}{9}

= - \frac{6}{9}

u = \frac{6}{9}, u = - \frac{6}{9}

代用を戻す

u = cos (x)

cos (x) = \frac{6}{9}, cos (x) = - \frac{6}{9}

cos (x) = \frac{6}{9}, cos (x) = - \frac{6}{9}

cos (x) = \frac{6}{9} : x = arccos (\frac{6}{9}) + 2 πn, x = 2 π - arccos (\frac{6}{9}) + 2 πn

cos (x) = \frac{6}{9}

三角関数の逆数プロパティを適用する

cos (x) = \frac{6}{9}

以下の一般解

cos (x) = \frac{6}{9}

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

x = arccos (\frac{6}{9}) + 2 πn, x = 2 π - arccos (\frac{6}{9}) + 2 πn

x = arccos (\frac{6}{9}) + 2 πn, x = 2 π - arccos (\frac{6}{9}) + 2 πn

cos (x) = - \frac{6}{9} : x = arccos (- \frac{6}{9}) + 2 πn, x = - arccos (- \frac{6}{9}) + 2 πn

cos (x) = - \frac{6}{9}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{6}{9}

以下の一般解

cos (x) = - \frac{6}{9}

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

x = arccos (- \frac{6}{9}) + 2 πn, x = - arccos (- \frac{6}{9}) + 2 πn

x = arccos (- \frac{6}{9}) + 2 πn, x = - arccos (- \frac{6}{9}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{6}{9}) + 2 πn, x = 2 π - arccos (\frac{6}{9}) + 2 πn, x = arccos (- \frac{6}{9}) + 2 πn, x = - arccos (- \frac{6}{9}) + 2 πn

10進法形式で解を証明する

x = 1.29515 \dots + 2 πn, x = 2 π - 1.29515 \dots + 2 πn, x = 1.84643 \dots + 2 πn, x = - 1.84643 \dots + 2 πn

グラフ

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