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人気のある三角関数 >

3sin(x)=2tan(x)

解

3 sin (x) = 2 tan (x)

解

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin (x) = 2 tan (x)

両辺から

2 tan (x)

を引く

3 sin (x) - 2 tan (x) = 0

サイン, コサインで表わす

- 2 tan (x) + 3 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 )}

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

簡素化

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

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

乗じる

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

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

分数を乗じる:

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

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

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

元を分数に変換する:

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

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

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

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

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

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

因数

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

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

共通項をくくり出す

sin (x)

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

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

各部分を別個に解く

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

3 cos (x) - 2 = 0 : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

3 cos (x) - 2 = 0

2

を右側に移動します

3 cos (x) - 2 = 0

両辺に

2

を足す

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

簡素化

3 cos (x) = 2

3 cos (x) = 2

以下で両辺を割る

3

3 cos (x) = 2

以下で両辺を割る

3

\frac{3 cos ( x )}{3} = \frac{2}{3}

簡素化

cos (x) = \frac{2}{3}

cos (x) = \frac{2}{3}

三角関数の逆数プロパティを適用する

cos (x) = \frac{2}{3}

以下の一般解

cos (x) = \frac{2}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

10進法形式で解を証明する

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

グラフ

人気の例

cos^2(x)+2cos(x)=0

cos^{2} (x) + 2 cos (x) = 0

2sin(x)=tan(2x)

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