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人気のある三角関数 >

tan(pi/4-x)+cot(pi/4-x)=4

解

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

解

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

+1

度

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

解答ステップ

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

三角関数の公式を使用して書き換える

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

三角関数の公式を使用して書き換える

cot (\frac{π}{4} - x)

基本的な三角関数の公式を使用する:

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

= \frac{cos ( \frac{π}{4} - x )}{sin ( \frac{π}{4} - x )}

角の差の公式を使用する:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{cos ( \frac{π}{4} - x )}{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}

角の差の公式を使用する:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

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

簡素化

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

\frac{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x)

簡素化

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

次の自明恒等式を使用する:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (\frac{π}{4}) sin (x)

簡素化

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

次の自明恒等式を使用する:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

簡素化

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

次の自明恒等式を使用する:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} cos (x) - cos (\frac{π}{4}) sin (x)

簡素化

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

次の自明恒等式を使用する:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

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

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

乗じる

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乗じる

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

分数を組み合わせる

\frac{2 cos ( x )}{2} - \frac{2 sin ( x )}{2} : \frac{2 cos ( x ) - 2 sin ( x )}{2}

規則を適用

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

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

= \frac{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

分数を組み合わせる

\frac{2 cos ( x )}{2} + \frac{2 sin ( x )}{2} : \frac{2 cos ( x ) + 2 sin ( x )}{2}

規則を適用

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

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

= \frac{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

分数を割る:

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

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

共通因数を約分する:

2

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

共通項をくくり出す

2

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

共通項をくくり出す

2

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

共通因数を約分する:

2

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

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

基本的な三角関数の公式を使用する:

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

= \frac{sin ( \frac{π}{4} - x )}{cos ( \frac{π}{4} - x )}

角の差の公式を使用する:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} - x )}

角の差の公式を使用する:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

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

簡素化

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

\frac{sin ( \frac{π}{4} ) cos ( x ) - cos ( \frac{π}{4} ) sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

簡素化

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

次の自明恒等式を使用する:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} cos (x) - cos (\frac{π}{4}) sin (x)

簡素化

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

次の自明恒等式を使用する:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x)

簡素化

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

次の自明恒等式を使用する:

cos (\frac{π}{4}) = \frac{2}{2}

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

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (\frac{π}{4}) sin (x)

簡素化

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

次の自明恒等式を使用する:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

乗じる

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乗じる

\frac{2}{2} cos (x) : \frac{2 cos ( x )}{2}

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

分数を乗じる:

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乗じる

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

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

分数を乗じる:

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

分数を組み合わせる

\frac{2 cos ( x )}{2} + \frac{2 sin ( x )}{2} : \frac{2 cos ( x ) + 2 sin ( x )}{2}

規則を適用

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

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

= \frac{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

分数を組み合わせる

\frac{2 cos ( x )}{2} - \frac{2 sin ( x )}{2} : \frac{2 cos ( x ) - 2 sin ( x )}{2}

規則を適用

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

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

= \frac{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

分数を割る:

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

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

共通因数を約分する:

2

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

共通項をくくり出す

2

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

共通項をくくり出す

2

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

共通因数を約分する:

2

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

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

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + \frac{cos ( x ) + sin ( x )}{cos ( x ) - sin ( x )} = 4

簡素化

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

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

以下の最小公倍数:

cos (x) + sin (x), cos (x) - sin (x) : (cos (x) + sin (x)) (cos (x) - sin (x))

cos (x) + sin (x), cos (x) - sin (x)

最小公倍数 (LCM)

cos (x) + sin (x)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (x) - sin (x)

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

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

(cos (x) + sin (x)) (cos (x) - sin (x))

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

の場合

:

分母と分子に以下を乗じる:

cos (x) - sin (x)

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

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

の場合

:

分母と分子に以下を乗じる:

cos (x) + sin (x)

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

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

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

拡張

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

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

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

完全平方式を適用する:

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

a = cos (x), b = sin (x)

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

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

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

完全平方式を適用する:

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

a = cos (x), b = sin (x)

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

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

簡素化

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

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

類似した元を足す:

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

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

類似した元を足す:

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

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

類似した元を足す:

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

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

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

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

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

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

両辺から

4

を引く

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

簡素化

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - 4 : \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

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

元を分数に変換する:

4 = \frac{4 ( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

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

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

拡張

2 cos^{2} (x) + 2 sin^{2} (x) - 4 (cos (x) + sin (x)) (cos (x) - sin (x)) : - 2 cos^{2} (x) + 6 sin^{2} (x)

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

拡張

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

拡張

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

(cos (x) + sin (x)) (cos (x) - sin (x))

2乗の差の公式を適用する:

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

a = cos (x), b = sin (x)

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

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

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

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

マイナス・プラスの規則を適用する

- (- a) = a

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

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

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

簡素化

2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x) : - 2 cos^{2} (x) + 6 sin^{2} (x)

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

類似した元を足す:

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

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

類似した元を足す:

2 sin^{2} (x) + 4 sin^{2} (x) = 6 sin^{2} (x)

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

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

= \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = 0

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

- 2 cos^{2} (x) + 6 sin^{2} (x) = 0

因数

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

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

6

を書き換え

3 \cdot 2

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

共通項をくくり出す

2

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

因数

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

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

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

を書き換え

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

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

累乗根の規則を適用する:

a = (a)^{2}

3 = (3)^{2}

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

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

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

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

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

2乗の差の公式を適用する:

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

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

= (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

= 2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x)) = 0

各部分を別個に解く

3 sin (x) + cos (x) = 0 or 3 sin (x) - cos (x) = 0

3 sin (x) + cos (x) = 0 : x = \frac{5 π}{6} + πn

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

三角関数の公式を使用して書き換える

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

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

で両辺を割る

\frac{3 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

簡素化

\frac{3 sin ( x )}{cos ( x )} + 1 = 0

基本的な三角関数の公式を使用する:

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

3 tan (x) + 1 = 0

3 tan (x) + 1 = 0

1

を右側に移動します

3 tan (x) + 1 = 0

両辺から

1

を引く

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

簡素化

3 tan (x) = - 1

3 tan (x) = - 1

以下で両辺を割る

3

3 tan (x) = - 1

以下で両辺を割る

3

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

簡素化

\frac{3 tan ( x )}{3} = \frac{- 1}{3}

簡素化

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

共通因数を約分する:

3

= tan (x)

簡素化

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

有理化する

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

共役で乗じる

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

累乗根の規則を適用する:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

tan (x) = - \frac{3}{3}

以下の一般解

tan (x) = - \frac{3}{3}

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

3 sin (x) - cos (x) = 0 : x = \frac{π}{6} + πn

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

三角関数の公式を使用して書き換える

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

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

で両辺を割る

\frac{3 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

簡素化

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

基本的な三角関数の公式を使用する:

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

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

1

を右側に移動します

3 tan (x) - 1 = 0

両辺に

1

を足す

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

簡素化

3 tan (x) = 1

3 tan (x) = 1

以下で両辺を割る

3

3 tan (x) = 1

以下で両辺を割る

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

簡素化

\frac{3 tan ( x )}{3} = \frac{1}{3}

簡素化

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

共通因数を約分する:

3

= tan (x)

簡素化

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

共役で乗じる

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

累乗根の規則を適用する:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

以下の一般解

tan (x) = \frac{3}{3}

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}{6} + πn

x = \frac{π}{6} + πn

すべての解を組み合わせる

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

グラフ

人気の例

7sin(θ)-1=5sin(θ)

7 sin (θ) - 1 = 5 sin (θ)

csc (θ) = - \frac{7}{3}

sin (θ) = \frac{5}{8}

arctan(2x)+arctan(x)= pi/4

arctan (2 x) + arctan (x) = \frac{π}{4}

tan (a) = 1