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人気のある三角関数 >

2cos(x)=3tan(x)

解

2 cos (x) = 3 tan (x)

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 cos (x) = 3 tan (x)

両辺から

3 tan (x)

を引く

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

サイン, コサインで表わす

2 cos (x) - 3 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 )}

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

簡素化

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

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

乗じる

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

3 \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 3}{cos ( x )}

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

元を分数に変換する:

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

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

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

2 cos (x) cos (x) - sin (x) \cdot 3

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

2 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)

= 2 cos^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 2 cos^{2} (x)

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

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

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

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

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

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

三角関数の公式を使用して書き換える

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

= 2 (1 - sin^{2} (x)) - 3 sin (x)

(1 - sin^{2} (x)) \cdot 2 - 3 sin (x) = 0

置換で解く

(1 - sin^{2} (x)) \cdot 2 - 3 sin (x) = 0

仮定:

sin (x) = u

(1 - u^{2}) \cdot 2 - 3 u = 0

(1 - u^{2}) \cdot 2 - 3 u = 0 : u = - 2, u = \frac{1}{2}

(1 - u^{2}) \cdot 2 - 3 u = 0

拡張

(1 - u^{2}) \cdot 2 - 3 u : 2 - 2 u^{2} - 3 u

(1 - u^{2}) \cdot 2 - 3 u

= 2 (1 - u^{2}) - 3 u

拡張

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

2 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

数を乗じる:

2 \cdot 1 = 2

= 2 - 2 u^{2}

= 2 - 2 u^{2} - 3 u

2 - 2 u^{2} - 3 u = 0

標準的な形式で書く

a x^{2} + b x + c = 0

- 2 u^{2} - 3 u + 2 = 0

解くとthe二次式

- 2 u^{2} - 3 u + 2 = 0

二次Equationの公式:

次の場合:

a = - 2, b = - 3, c = 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

(- 3)^{2} - 4 (- 2) \cdot 2 = 5

(- 3)^{2} - 4 (- 2) \cdot 2

規則を適用

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 2

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 2 \cdot 2

数を乗じる:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数を足す:

9 + 16 = 25

= 25

数を因数に分解する:

25 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 ( - 2 )}

解を分離する

u_{1} = \frac{- ( - 3 ) + 5}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 5}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 5}{2 ( - 2 )} : - 2

\frac{- ( - 3 ) + 5}{2 ( - 2 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{3 + 5}{- 2 \cdot 2}

数を足す:

3 + 5 = 8

= \frac{8}{- 2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{8}{- 4}

分数の規則を適用する:

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

= - \frac{8}{4}

数を割る:

\frac{8}{4} = 2

= - 2

u = \frac{- ( - 3 ) - 5}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 3 ) - 5}{2 ( - 2 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{3 - 5}{- 2 \cdot 2}

数を引く:

3 - 5 = - 2

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

数を乗じる:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

分数の規則を適用する:

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

= \frac{2}{4}

共通因数を約分する:

2

= \frac{1}{2}

二次equationの解:

u = - 2, u = \frac{1}{2}

代用を戻す

u = sin (x)

sin (x) = - 2, sin (x) = \frac{1}{2}

sin (x) = - 2, sin (x) = \frac{1}{2}

sin (x) = - 2 :

解なし

sin (x) = - 2

- 1 \leq sin (x) \leq 1

解なし

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

以下の一般解

sin (x) = \frac{1}{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}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

人気の例

cos (θ) = \frac{1}{4}

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

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

sin (θ) = \frac{1}{3}

cos(θ)= 1/(sqrt(2))

cos (θ) = \frac{1}{2}

tan(2x)=3tan(x)

tan (2 x) = 3 tan (x)