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人気のある三角関数 >

csc(x)cos^2(x)+sin^2(x)=csc(x)

解

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

解

x = \frac{π}{2} + 2 πn

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

csc (x)

を引く

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

三角関数の公式を使用して書き換える

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

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

簡素化

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

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

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

拡張

csc (x) (1 - sin^{2} (x)) : csc (x) - sin^{2} (x) csc (x)

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

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

乗算:

1 \cdot csc (x) = csc (x)

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

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

類似した元を足す:

- csc (x) + csc (x) = 0

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

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

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

因数

sin^{2} (x) - csc (x) sin^{2} (x) : - sin^{2} (x) (csc (x) - 1)

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

共通項をくくり出す

- sin^{2} (x)

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

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

各部分を別個に解く

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

規則を適用

x^{n} = 0 \Rightarrow x = 0

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

csc (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

csc (x) - 1 = 0

1

を右側に移動します

csc (x) - 1 = 0

両辺に

1

を足す

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

簡素化

csc (x) = 1

csc (x) = 1

以下の一般解

csc (x) = 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn

equationは以下で未定義のため:

2 πn, π + 2 πn

x = \frac{π}{2} + 2 πn

グラフ

人気の例

cos((2pix)/3)= 1/2

cos (\frac{2 π x}{3}) = \frac{1}{2}

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

-sqrt(1+tan(x))=sec(x)

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

sin(x/2)=sin(2x)

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

cos (x) = cos (1)