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人気のある三角関数 >

sin^4(x)+cos^4(x)= 7/9

解

sin^{4} (x) + cos^{4} (x) = \frac{7}{9}

解

x = 0.36486 \dots + 2 πn, x = 2 π - 0.36486 \dots + 2 πn, x = 2.77672 \dots + 2 πn, x = - 2.77672 \dots + 2 πn, x = 1.20593 \dots + 2 πn, x = 2 π - 1.20593 \dots + 2 πn, x = 1.93566 \dots + 2 πn, x = - 1.93566 \dots + 2 πn

+1

度

x = 20.90515 \dots^{\circ} + 36 0^{\circ} n, x = 339.09484 \dots^{\circ} + 36 0^{\circ} n, x = 159.09484 \dots^{\circ} + 36 0^{\circ} n, x = - 159.09484 \dots^{\circ} + 36 0^{\circ} n, x = 69.09484 \dots^{\circ} + 36 0^{\circ} n, x = 290.90515 \dots^{\circ} + 36 0^{\circ} n, x = 110.90515 \dots^{\circ} + 36 0^{\circ} n, x = - 110.90515 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{4} (x) + cos^{4} (x) = \frac{7}{9}

両辺から

\frac{7}{9}

を引く

sin^{4} (x) + cos^{4} (x) - \frac{7}{9} = 0

簡素化

sin^{4} (x) + cos^{4} (x) - \frac{7}{9} : \frac{9 sin ^{4} ( x ) + 9 cos ^{4} ( x ) - 7}{9}

sin^{4} (x) + cos^{4} (x) - \frac{7}{9}

元を分数に変換する:

sin^{4} (x) = \frac{sin ^{4} ( x ) 9}{9}, cos^{4} (x) = \frac{cos ^{4} ( x ) 9}{9}

= \frac{sin ^{4} ( x ) \cdot 9}{9} + \frac{cos ^{4} ( x ) \cdot 9}{9} - \frac{7}{9}

分母が等しいので, 分数を組み合わせる:

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

= \frac{sin ^{4} ( x ) \cdot 9 + cos ^{4} ( x ) \cdot 9 - 7}{9}

\frac{9 sin ^{4} ( x ) + 9 cos ^{4} ( x ) - 7}{9} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

9 sin^{4} (x) + 9 cos^{4} (x) - 7 = 0

指数の規則を適用する:

a^{b} = a^{2} a^{b - 2}

- 7 + 9 cos^{4} (x) + 9 sin^{2} (x) sin^{2} (x) = 0

三角関数の公式を使用して書き換える

- 7 + 9 cos^{4} (x) + 9 sin^{2} (x) sin^{2} (x)

ピタゴラスの公式を使用する:

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

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

= - 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

簡素化

- 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x)) : 18 cos^{4} (x) - 18 cos^{2} (x) + 2

- 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

9 (1 - cos^{2} (x)) (1 - cos^{2} (x)) = 9 (1 - cos^{2} (x))^{2}

9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

指数の規則を適用する:

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

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

= 9 (1 - cos^{2} (x))^{1 + 1}

数を足す:

1 + 1 = 2

= 9 (1 - cos^{2} (x))^{2}

= - 7 + 9 cos^{4} (x) + 9 (- cos^{2} (x) + 1)^{2}

(1 - cos^{2} (x))^{2} : 1 - 2 cos^{2} (x) + cos^{4} (x)

完全平方式を適用する:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = cos^{2} (x)

= 1^{2} - 2 \cdot 1 \cdot cos^{2} (x) + (cos^{2} (x))^{2}

簡素化

1^{2} - 2 \cdot 1 \cdot cos^{2} (x) + (cos^{2} (x))^{2} : 1 - 2 cos^{2} (x) + cos^{4} (x)

1^{2} - 2 \cdot 1 \cdot cos^{2} (x) + (cos^{2} (x))^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 2 \cdot 1 \cdot cos^{2} (x) + (cos^{2} (x))^{2}

2 \cdot 1 \cdot cos^{2} (x) = 2 cos^{2} (x)

2 \cdot 1 \cdot cos^{2} (x)

数を乗じる:

2 \cdot 1 = 2

= 2 cos^{2} (x)

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

(cos^{2} (x))^{2}

指数の規則を適用する:

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

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

数を乗じる:

2 \cdot 2 = 4

= cos^{4} (x)

= 1 - 2 cos^{2} (x) + cos^{4} (x)

= 1 - 2 cos^{2} (x) + cos^{4} (x)

= - 7 + 9 cos^{4} (x) + 9 (1 - 2 cos^{2} (x) + cos^{4} (x))

拡張

9 (1 - 2 cos^{2} (x) + cos^{4} (x)) : 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

9 (1 - 2 cos^{2} (x) + cos^{4} (x))

括弧を分配する

= 9 \cdot 1 + 9 (- 2 cos^{2} (x)) + 9 cos^{4} (x)

マイナス・プラスの規則を適用する

+ (- a) = - a

= 9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

簡素化

9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x) : 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

数を乗じる:

9 \cdot 1 = 9

= 9 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

数を乗じる:

9 \cdot 2 = 18

= 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

= 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

= - 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

簡素化

- 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x) : 18 cos^{4} (x) - 18 cos^{2} (x) + 2

- 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

条件のようなグループ

= 9 cos^{4} (x) - 18 cos^{2} (x) + 9 cos^{4} (x) - 7 + 9

類似した元を足す:

9 cos^{4} (x) + 9 cos^{4} (x) = 18 cos^{4} (x)

= 18 cos^{4} (x) - 18 cos^{2} (x) - 7 + 9

数を足す/引く:

- 7 + 9 = 2

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

2 - 18 cos^{2} (x) + 18 cos^{4} (x) = 0

置換で解く

2 - 18 cos^{2} (x) + 18 cos^{4} (x) = 0

仮定:

cos (x) = u

2 - 18 u^{2} + 18 u^{4} = 0

2 - 18 u^{2} + 18 u^{4} = 0 : u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}, u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

2 - 18 u^{2} + 18 u^{4} = 0

標準的な形式で書く

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

18 u^{4} - 18 u^{2} + 2 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

18 v^{2} - 18 v + 2 = 0

解く

18 v^{2} - 18 v + 2 = 0 : v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

18 v^{2} - 18 v + 2 = 0

解くとthe二次式

18 v^{2} - 18 v + 2 = 0

二次Equationの公式:

次の場合:

a = 18, b = - 18, c = 2

v_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 18 \cdot 2}{2 \cdot 18}

v_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 18 \cdot 2}{2 \cdot 18}

(- 18)^{2} - 4 \cdot 18 \cdot 2 = 65

(- 18)^{2} - 4 \cdot 18 \cdot 2

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 18)^{2} = 1 8^{2}

= 1 8^{2} - 4 \cdot 18 \cdot 2

数を乗じる:

4 \cdot 18 \cdot 2 = 144

= 1 8^{2} - 144

1 8^{2} = 324

= 324 - 144

数を引く:

324 - 144 = 180

= 180

以下の素因数分解:

180 : 2^{2} \cdot 3^{2} \cdot 5

180

1802180 = 90 \cdot 2

で割る

= 2 \cdot 90

90290 = 45 \cdot 2

で割る

= 2 \cdot 2 \cdot 45

45345 = 15 \cdot 3

で割る

= 2 \cdot 2 \cdot 3 \cdot 15

15315 = 5 \cdot 3

で割る

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

2, 3, 5

はすべて素数である。ゆえにさらに因数分解することはできない

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

= 2^{2} \cdot 3^{2} \cdot 5

= 2^{2} \cdot 3^{2} \cdot 5

累乗根の規則を適用する:

n ab = n a n b

= 5 2^{2} 3^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 25 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 2 \cdot 35

改良

= 65

v_{1, 2} = \frac{- ( - 18 ) \pm 6 5}{2 \cdot 18}

解を分離する

v_{1} = \frac{- ( - 18 ) + 6 5}{2 \cdot 18}, v_{2} = \frac{- ( - 18 ) - 6 5}{2 \cdot 18}

v = \frac{- ( - 18 ) + 6 5}{2 \cdot 18} : \frac{3 + 5}{6}

\frac{- ( - 18 ) + 6 5}{2 \cdot 18}

規則を適用

- (- a) = a

= \frac{18 + 6 5}{2 \cdot 18}

数を乗じる:

2 \cdot 18 = 36

= \frac{18 + 6 5}{36}

因数

18 + 65 : 6 (3 + 5)

18 + 65

書き換え

= 6 \cdot 3 + 65

共通項をくくり出す

6

= 6 (3 + 5)

= \frac{6 ( 3 + 5 )}{36}

共通因数を約分する:

6

= \frac{3 + 5}{6}

v = \frac{- ( - 18 ) - 6 5}{2 \cdot 18} : \frac{3 - 5}{6}

\frac{- ( - 18 ) - 6 5}{2 \cdot 18}

規則を適用

- (- a) = a

= \frac{18 - 6 5}{2 \cdot 18}

数を乗じる:

2 \cdot 18 = 36

= \frac{18 - 6 5}{36}

因数

18 - 65 : 6 (3 - 5)

18 - 65

書き換え

= 6 \cdot 3 - 65

共通項をくくり出す

6

= 6 (3 - 5)

= \frac{6 ( 3 - 5 )}{36}

共通因数を約分する:

6

= \frac{3 - 5}{6}

二次equationの解:

v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = \frac{3 + 5}{6} : u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}

u^{2} = \frac{3 + 5}{6}

x^{2} = f (a)

の場合, 解は

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

u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}

解く

u^{2} = \frac{3 - 5}{6} : u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

u^{2} = \frac{3 - 5}{6}

x^{2} = f (a)

の場合, 解は

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

u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

解答は

u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}, u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

代用を戻す

u = cos (x)

cos (x) = \frac{3 + 5}{6}, cos (x) = - \frac{3 + 5}{6}, cos (x) = \frac{3 - 5}{6}, cos (x) = - \frac{3 - 5}{6}

cos (x) = \frac{3 + 5}{6}, cos (x) = - \frac{3 + 5}{6}, cos (x) = \frac{3 - 5}{6}, cos (x) = - \frac{3 - 5}{6}

cos (x) = \frac{3 + 5}{6} : x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

cos (x) = \frac{3 + 5}{6}

三角関数の逆数プロパティを適用する

cos (x) = \frac{3 + 5}{6}

以下の一般解

cos (x) = \frac{3 + 5}{6}

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

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

cos (x) = - \frac{3 + 5}{6} : x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

cos (x) = - \frac{3 + 5}{6}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{3 + 5}{6}

以下の一般解

cos (x) = - \frac{3 + 5}{6}

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

x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

cos (x) = \frac{3 - 5}{6} : x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

cos (x) = \frac{3 - 5}{6}

三角関数の逆数プロパティを適用する

cos (x) = \frac{3 - 5}{6}

以下の一般解

cos (x) = \frac{3 - 5}{6}

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

x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

cos (x) = - \frac{3 - 5}{6} : x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

cos (x) = - \frac{3 - 5}{6}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{3 - 5}{6}

以下の一般解

cos (x) = - \frac{3 - 5}{6}

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

x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

すべての解を組み合わせる

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn, x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn, x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn, x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

10進法形式で解を証明する

x = 0.36486 \dots + 2 πn, x = 2 π - 0.36486 \dots + 2 πn, x = 2.77672 \dots + 2 πn, x = - 2.77672 \dots + 2 πn, x = 1.20593 \dots + 2 πn, x = 2 π - 1.20593 \dots + 2 πn, x = 1.93566 \dots + 2 πn, x = - 1.93566 \dots + 2 πn

グラフ

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