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人気のある三角関数 >

sin(x)=tan(x/2)

解

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

解

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

+1

度

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 27 0^{\circ} + 72 0^{\circ} n, x = 45 0^{\circ} + 72 0^{\circ} n, x = 9 0^{\circ} + 72 0^{\circ} n, x = 63 0^{\circ} + 72 0^{\circ} n

解答ステップ

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

両辺から

tan (\frac{x}{2})

を引く

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

仮定:

u = \frac{x}{2}

sin (2 u) - tan (u) = 0

サイン, コサインで表わす

sin (2 u) - tan (u)

基本的な三角関数の公式を使用する:

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

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

簡素化

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

sin (2 u) - \frac{sin ( u )}{cos ( u )}

元を分数に変換する:

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

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

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 u ) cos ( u ) - sin ( u )}{cos ( u )}

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

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

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

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

三角関数の公式を使用して書き換える

- sin (u) + cos (u) sin (2 u)

2倍角の公式を使用:

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

= - sin (u) + cos (u) \cdot 2 sin (u) cos (u)

cos (u) \cdot 2 sin (u) cos (u) = 2 cos^{2} (u) sin (u)

cos (u) \cdot 2 sin (u) cos (u)

指数の規則を適用する:

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

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

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

数を足す:

1 + 1 = 2

= 2 sin (u) cos^{2} (u)

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

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

因数

- sin (u) + 2 cos^{2} (u) sin (u) : sin (u) (2 cos (u) + 1) (2 cos (u) - 1)

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

共通項をくくり出す

sin (u)

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

因数

2 cos^{2} (u) - 1 : (2 cos (u) + 1) (2 cos (u) - 1)

2 cos^{2} (u) - 1

2 cos^{2} (u) - 1

を書き換え

(2 cos (u))^{2} - 1^{2}

2 cos^{2} (u) - 1

累乗根の規則を適用する:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (u) - 1

1

を書き換え

1^{2}

= (2)^{2} cos^{2} (u) - 1^{2}

指数の規則を適用する:

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

(2)^{2} cos^{2} (u) = (2 cos (u))^{2}

= (2 cos (u))^{2} - 1^{2}

= (2 cos (u))^{2} - 1^{2}

2乗の差の公式を適用する:

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

(2 cos (u))^{2} - 1^{2} = (2 cos (u) + 1) (2 cos (u) - 1)

= (2 cos (u) + 1) (2 cos (u) - 1)

= sin (u) (2 cos (u) + 1) (2 cos (u) - 1)

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

各部分を別個に解く

sin (u) = 0 or 2 cos (u) + 1 = 0 or 2 cos (u) - 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

以下の一般解

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

解く

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) + 1 = 0 : u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

2 cos (u) + 1 = 0

1

を右側に移動します

2 cos (u) + 1 = 0

両辺から

1

を引く

2 cos (u) + 1 - 1 = 0 - 1

簡素化

2 cos (u) = - 1

2 cos (u) = - 1

以下で両辺を割る

2

2 cos (u) = - 1

以下で両辺を割る

2

\frac{2 cos ( u )}{2} = \frac{- 1}{2}

簡素化

\frac{2 cos ( u )}{2} = \frac{- 1}{2}

簡素化

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

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

共通因数を約分する:

2

= cos (u)

簡素化

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

分数の規則を適用する:

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

= - \frac{1}{2}

有理化する

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

共役で乗じる

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (u) = - \frac{2}{2}

cos (u) = - \frac{2}{2}

cos (u) = - \frac{2}{2}

以下の一般解

cos (u) = - \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}

u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn

2 cos (u) - 1 = 0 : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

2 cos (u) - 1 = 0

1

を右側に移動します

2 cos (u) - 1 = 0

両辺に

1

を足す

2 cos (u) - 1 + 1 = 0 + 1

簡素化

2 cos (u) = 1

2 cos (u) = 1

以下で両辺を割る

2

2 cos (u) = 1

以下で両辺を割る

2

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

簡素化

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

簡素化

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

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

共通因数を約分する:

2

= cos (u)

簡素化

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

共役で乗じる

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

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

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

以下の一般解

cos (u) = \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}

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

すべての解を組み合わせる

u = 2 πn, u = π + 2 πn, u = \frac{3 π}{4} + 2 πn, u = \frac{5 π}{4} + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

代用を戻す

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot 2 πn

簡素化

x = 4 πn

x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = π + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

簡素化

x = 2 π + 4 πn

x = 2 π + 4 πn

\frac{x}{2} = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{2} + 4 πn

\frac{x}{2} = \frac{3 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{3 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn : \frac{3 π}{2} + 4 πn

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

分数を乗じる:

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

= \frac{3 π 2}{4}

数を乗じる:

3 \cdot 2 = 6

= \frac{6 π}{4}

共通因数を約分する:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

\frac{x}{2} = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{2} + 4 πn

\frac{x}{2} = \frac{5 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{5 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn : \frac{5 π}{2} + 4 πn

2 \cdot \frac{5 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{4} = \frac{5 π}{2}

2 \cdot \frac{5 π}{4}

分数を乗じる:

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

= \frac{5 π 2}{4}

数を乗じる:

5 \cdot 2 = 10

= \frac{10 π}{4}

共通因数を約分する:

2

= \frac{5 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

x = \frac{5 π}{2} + 4 πn

\frac{x}{2} = \frac{π}{4} + 2 πn : x = \frac{π}{2} + 4 πn

\frac{x}{2} = \frac{π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{π}{4} + 2 \cdot 2 πn : \frac{π}{2} + 4 πn

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

分数を乗じる:

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

= \frac{π 2}{4}

共通因数を約分する:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

\frac{x}{2} = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{2} + 4 πn

\frac{x}{2} = \frac{7 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{7 π}{4} + 2 πn

以下で両辺を乗じる:

2

\frac{2 x}{2} = 2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} = 2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn : \frac{7 π}{2} + 4 πn

2 \cdot \frac{7 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{4} = \frac{7 π}{2}

2 \cdot \frac{7 π}{4}

分数を乗じる:

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

= \frac{7 π 2}{4}

数を乗じる:

7 \cdot 2 = 14

= \frac{14 π}{4}

共通因数を約分する:

2

= \frac{7 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = \frac{7 π}{2} + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = \frac{3 π}{2} + 4 πn, x = \frac{5 π}{2} + 4 πn, x = \frac{π}{2} + 4 πn, x = \frac{7 π}{2} + 4 πn

グラフ

人気の例

2sin^2(x)-17sin(x)+8=0

2 sin^{2} (x) - 17 sin (x) + 8 = 0

4cos^2(t)sin(t)-2cos(t)sin(t)=0

4 cos^{2} (t) sin (t) - 2 cos (t) sin (t) = 0

3(1-sin(θ))=2cos^2(θ)

3 (1 - sin (θ)) = 2 cos^{2} (θ)

sin(x)=(sqrt(5))/5

sin (x) = \frac{5}{5}

tan(θ)=0.11246

tan (θ) = 0.11246