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人気のある三角関数 >

sin^6(x)-cos^6(x)=0

解

sin^{6} (x) - cos^{6} (x) = 0

解

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

+1

度

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

解答ステップ

sin^{6} (x) - cos^{6} (x) = 0

因数

sin^{6} (x) - cos^{6} (x) : (sin (x) + cos (x)) (sin^{2} (x) - sin (x) cos (x) + cos^{2} (x)) (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

sin^{6} (x) - cos^{6} (x)

sin^{6} (x) - cos^{6} (x)

を書き換え

(sin^{3} (x))^{2} - (cos^{3} (x))^{2}

sin^{6} (x) - cos^{6} (x)

指数の規則を適用する:

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

sin^{6} (x) = (sin^{3} (x))^{2}

= (sin^{3} (x))^{2} - cos^{6} (x)

指数の規則を適用する:

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

cos^{6} (x) = (cos^{3} (x))^{2}

= (sin^{3} (x))^{2} - (cos^{3} (x))^{2}

= (sin^{3} (x))^{2} - (cos^{3} (x))^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(sin^{3} (x))^{2} - (cos^{3} (x))^{2} = (sin^{3} (x) + cos^{3} (x)) (sin^{3} (x) - cos^{3} (x))

= (sin^{3} (x) + cos^{3} (x)) (sin^{3} (x) - cos^{3} (x))

因数

sin^{3} (x) + cos^{3} (x) : (sin (x) + cos (x)) (sin^{2} (x) - sin (x) cos (x) + cos^{2} (x))

sin^{3} (x) + cos^{3} (x)

立方数の和の公式を適用する:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

sin^{3} (x) + cos^{3} (x) = (sin (x) + cos (x)) (sin^{2} (x) - sin (x) cos (x) + cos^{2} (x))

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

= (sin (x) + cos (x)) (sin^{2} (x) - sin (x) cos (x) + cos^{2} (x)) (sin^{3} (x) - cos^{3} (x))

因数

sin^{3} (x) - cos^{3} (x) : (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

sin^{3} (x) - cos^{3} (x)

立方数の差の公式を適用する:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

sin^{3} (x) - cos^{3} (x) = (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

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

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

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

三角関数の公式を使用して書き換える

(sin (x) + cos (x)) (sin^{2} (x) - sin (x) cos (x) + cos^{2} (x)) (sin (x) - cos (x)) (sin^{2} (x) + sin (x) cos (x) + cos^{2} (x))

ピタゴラスの公式を使用する:

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

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

(- cos (x) + sin (x)) (cos (x) + sin (x)) (cos (x) sin (x) + 1) (cos^{2} (x) + sin^{2} (x) - cos (x) sin (x)) = 0

各部分を別個に解く

- cos (x) + sin (x) = 0 or cos (x) + sin (x) = 0 or cos (x) sin (x) + 1 = 0 or cos^{2} (x) + sin^{2} (x) - cos (x) sin (x) = 0

- cos (x) + sin (x) = 0 : x = \frac{π}{4} + πn

- cos (x) + sin (x) = 0

三角関数の公式を使用して書き換える

- cos (x) + sin (x) = 0

cos (x), cos (x) \neq = 0

で両辺を割る

\frac{- cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

簡素化

- 1 + \frac{sin ( x )}{cos ( x )} = 0

基本的な三角関数の公式を使用する:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 1 + tan (x) = 0

- 1 + tan (x) = 0

1

を右側に移動します

- 1 + tan (x) = 0

両辺に

1

を足す

- 1 + tan (x) + 1 = 0 + 1

簡素化

tan (x) = 1

tan (x) = 1

以下の一般解

tan (x) = 1

tan (x)

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

三角関数の公式を使用して書き換える

cos (x) + sin (x) = 0

cos (x), cos (x) \neq = 0

で両辺を割る

\frac{cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

簡素化

1 + \frac{sin ( x )}{cos ( x )} = 0

基本的な三角関数の公式を使用する:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 + tan (x) = 0

1 + tan (x) = 0

1

を右側に移動します

1 + tan (x) = 0

両辺から

1

を引く

1 + tan (x) - 1 = 0 - 1

簡素化

tan (x) = - 1

tan (x) = - 1

以下の一般解

tan (x) = - 1

tan (x)

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

cos (x) sin (x) + 1 = 0 :

解なし

cos (x) sin (x) + 1 = 0

三角関数の公式を使用して書き換える

cos (x) sin (x) + 1

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

= 1 + \frac{sin ( 2 x )}{2}

1 + \frac{sin ( 2 x )}{2} = 0

1

を右側に移動します

1 + \frac{sin ( 2 x )}{2} = 0

両辺から

1

を引く

1 + \frac{sin ( 2 x )}{2} - 1 = 0 - 1

簡素化

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

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

以下で両辺を乗じる:

2

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

以下で両辺を乗じる:

2

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

簡素化

sin (2 x) = - 2

sin (2 x) = - 2

- 1 \leq sin (x) \leq 1

解なし

cos^{2} (x) + sin^{2} (x) - cos (x) sin (x) = 0 :

解なし

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

三角関数の公式を使用して書き換える

- cos (x) sin (x) + 1

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

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

1 - \frac{sin ( 2 x )}{2} = 0

1

を右側に移動します

1 - \frac{sin ( 2 x )}{2} = 0

両辺から

1

を引く

1 - \frac{sin ( 2 x )}{2} - 1 = 0 - 1

簡素化

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

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

以下で両辺を乗じる:

2

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

以下で両辺を乗じる:

2

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

簡素化

- sin (2 x) = - 2

- sin (2 x) = - 2

以下で両辺を割る

- 1

- sin (2 x) = - 2

以下で両辺を割る

- 1

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

簡素化

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

解なし

すべての解を組み合わせる

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

グラフ

人気の例

-sin(2x)sin(x)+cos(2x)cos(x)=0

- sin (2 x) sin (x) + cos (2 x) cos (x) = 0

sin(2x)+cos(2x)-1=0

sin (2 x) + cos (2 x) - 1 = 0

4tan(x)sin^2(x)+3=4sin^2(x)+3tan(x)

4 tan (x) sin^{2} (x) + 3 = 4 sin^{2} (x) + 3 tan (x)

tan (a) = 0.384

cos (b) = \frac{38}{70}