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人気のある三角関数 >

tan^2(x)+cos^2(x)-1=0

解

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

解

x = 2 πn, x = π + 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

三角関数の公式を使用して書き換える

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

基本的な三角関数の公式を使用する:

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

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

指数の規則を適用する:

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

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

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

簡素化

- 1 + 1 - sin^{2} (x) + \frac{sin ^{2} ( x )}{1 - sin ^{2} ( x )} : \frac{sin ^{4} ( x )}{- sin ^{2} ( x ) + 1}

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

- 1 + 1 = 0

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

元を分数に変換する:

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

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

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

拡張

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

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

拡張

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

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

分配法則を適用する:

a (b + c) = ab + a c

a = - sin^{2} (x), b = - sin^{2} (x), c = 1

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

マイナス・プラスの規則を適用する

- (- a) = a, + (- a) = - a

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

簡素化

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

sin^{2} (x) sin^{2} (x) - 1 \cdot sin^{2} (x)

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

sin^{2} (x) sin^{2} (x)

指数の規則を適用する:

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

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

= sin^{2 + 2} (x)

数を足す:

2 + 2 = 4

= sin^{4} (x)

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

1 \cdot sin^{2} (x)

乗算:

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

= sin^{2} (x)

= sin^{4} (x) - sin^{2} (x)

= sin^{4} (x) - sin^{2} (x)

= sin^{2} (x) + sin^{4} (x) - sin^{2} (x)

類似した元を足す:

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

= sin^{4} (x)

= \frac{sin ^{4} ( x )}{- sin ^{2} ( x ) + 1}

= \frac{sin ^{4} ( x )}{- sin ^{2} ( x ) + 1}

\frac{sin ^{4} ( x )}{1 - sin ^{2} ( x )} = 0

置換で解く

\frac{sin ^{4} ( x )}{1 - sin ^{2} ( x )} = 0

仮定:

sin (x) = u

\frac{u ^{4}}{1 - u ^{2}} = 0

\frac{u ^{4}}{1 - u ^{2}} = 0 : u = 0

\frac{u ^{4}}{1 - u ^{2}} = 0

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

u^{4} = 0

解く

u^{4} = 0 : u = 0

u^{4} = 0

零因子の原則を使用:

ab = 0

ならば

a = 0

または

b = 0

u = 0

解は

u = 0

u = 0

解を検算する

未定義の (特異) 点を求める:

u = 1, u = - 1

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

の分母をゼロに比較する

解く

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

1

を右側に移動します

1 - u^{2} = 0

両辺から

1

を引く

1 - u^{2} - 1 = 0 - 1

簡素化

- u^{2} = - 1

- u^{2} = - 1

以下で両辺を割る

- 1

- u^{2} = - 1

以下で両辺を割る

- 1

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

簡素化

u^{2} = 1

u^{2} = 1

x^{2} = f (a)

の場合, 解は

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

u = 1, u = - 1

1 = 1

1

累乗根の規則を適用する:

1 = 1

= 1

- 1 = - 1

- 1

累乗根の規則を適用する:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

以下の点は定義されていない

u = 1, u = - 1

未定義のポイントを解に組み合わせる:

u = 0

代用を戻す

u = sin (x)

sin (x) = 0

sin (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

以下の一般解

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn

グラフ

人気の例

cot^3(x)+cot(x)=0

21+18cos(x)=16(1-cos^2(x))

sin(2a+10)=cos(3a-20)

b= 3/((cot(x)))

4sin^2(x)+cos(x)+1=0