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人気のある三角関数 >

tan(x)+tan(x+45)=-2

解

tan (x) + tan (x + 4 5^{\circ}) = - 2

解

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

+1

ラジアン

x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

解答ステップ

tan (x) + tan (x + 4 5^{\circ}) = - 2

三角関数の公式を使用して書き換える

tan (x) + tan (x + 4 5^{\circ}) = - 2

三角関数の公式を使用して書き換える

tan (x + 4 5^{\circ})

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 ( x + 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

角の和の公式を使用する:

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

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

角の和の公式を使用する:

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

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

簡素化

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )} : \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ}) = \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ})

簡素化

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

次の自明恒等式を使用する:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

36 0^{\circ} n

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} sin (x) + sin (4 5^{\circ}) cos (x)

簡素化

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

次の自明恒等式を使用する:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

36 0^{\circ} n

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ}) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ})

簡素化

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

次の自明恒等式を使用する:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

36 0^{\circ} n

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 (4 5^{\circ}) sin (x)

簡素化

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

次の自明恒等式を使用する:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

36 0^{\circ} n

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

乗じる

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

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

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

乗じる

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

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

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

乗じる

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

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

分数を乗じる:

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

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

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

乗じる

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

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

分数を乗じる:

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

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

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

規則を適用

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

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

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

共通因数を約分する:

2

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

共通項をくくり出す

2

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

共通項をくくり出す

2

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

共通因数を約分する:

2

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

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

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

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

両辺から

- 2

を引く

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

簡素化

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} + 2 : \frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )}

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

元を分数に変換する:

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

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

拡張

tan (x) (cos (x) - sin (x)) + sin (x) + cos (x) + 2 (cos (x) - sin (x)) : tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

tan (x) (cos (x) - sin (x)) + sin (x) + cos (x) + 2 (cos (x) - sin (x))

拡張

tan (x) (cos (x) - sin (x)) : tan (x) cos (x) - tan (x) sin (x)

tan (x) (cos (x) - sin (x))

分配法則を適用する:

a (b - c) = ab - a c

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

= tan (x) cos (x) - tan (x) sin (x)

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

拡張

2 (cos (x) - sin (x)) : 2 cos (x) - 2 sin (x)

2 (cos (x) - sin (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = cos (x), c = sin (x)

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

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

簡素化

tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 cos (x) - 2 sin (x) : tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 cos (x) - 2 sin (x)

類似した元を足す:

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

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

類似した元を足す:

sin (x) - 2 sin (x) = - sin (x)

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

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

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

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

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

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

サイン, コサインで表わす

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

基本的な三角関数の公式を使用する:

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

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

簡素化

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

- sin (x) + 3 cos (x) + cos (x) \frac{sin ( x )}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

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

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

分数を乗じる:

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

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

共通因数を約分する:

cos (x)

= sin (x)

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

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

分数を乗じる:

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

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

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

sin (x) sin (x)

指数の規則を適用する:

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

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

= sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= sin^{2} (x)

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

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

類似した元を足す:

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

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

元を分数に変換する:

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

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

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

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

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

3 cos (x) cos (x)

指数の規則を適用する:

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

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

= 3 cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 3 cos^{2} (x)

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

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

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

\frac{- sin ^{2} ( x ) + 3 cos ^{2} ( x )}{cos ( x )} = 0

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

- sin^{2} (x) + 3 cos^{2} (x) = 0

因数

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

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

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

を書き換え

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

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

累乗根の規則を適用する:

a = (a)^{2}

3 = (3)^{2}

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

指数の規則を適用する:

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

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

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

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

2乗の差の公式を適用する:

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

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

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

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

各部分を別個に解く

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

3 cos (x) + sin (x) = 0 : x = 12 0^{\circ} + 18 0^{\circ} n

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

三角関数の公式を使用して書き換える

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

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

で両辺を割る

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

簡素化

3 + \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)

3 + tan (x) = 0

3 + tan (x) = 0

3

を右側に移動します

3 + tan (x) = 0

両辺から

3

を引く

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

簡素化

tan (x) = - 3

tan (x) = - 3

以下の一般解

tan (x) = - 3

tan (x)

18 0^{\circ} n

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 = 12 0^{\circ} + 18 0^{\circ} n

x = 12 0^{\circ} + 18 0^{\circ} n

3 cos (x) - sin (x) = 0 : x = 6 0^{\circ} + 18 0^{\circ} n

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

三角関数の公式を使用して書き換える

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

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

で両辺を割る

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

簡素化

3 - \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)

3 - tan (x) = 0

3 - tan (x) = 0

3

を右側に移動します

3 - tan (x) = 0

両辺から

3

を引く

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

簡素化

- tan (x) = - 3

- tan (x) = - 3

以下で両辺を割る

- 1

- tan (x) = - 3

以下で両辺を割る

- 1

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

簡素化

tan (x) = 3

tan (x) = 3

以下の一般解

tan (x) = 3

tan (x)

18 0^{\circ} n

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 = 6 0^{\circ} + 18 0^{\circ} n

x = 6 0^{\circ} + 18 0^{\circ} n

すべての解を組み合わせる

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

グラフ

人気の例

tan(x)=(22.5)/(34)

tan (x) = \frac{22.5}{34}

tan^{2} (4 x) = 0

tan(x-45)-tan(x+45)=4

tan (x - 4 5^{\circ}) - tan (x + 4 5^{\circ}) = 4

sin(2x)=10cos(x)

sin (2 x) = 10 cos (x)

3-4sin(θ)=4-6sin(θ)

3 - 4 sin (θ) = 4 - 6 sin (θ)