ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある三角関数 >

3cos(x)=8tan(x)

解

3 cos (x) = 8 tan (x)

解

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

+1

度

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 cos (x) = 8 tan (x)

両辺から

8 tan (x)

を引く

3 cos (x) - 8 tan (x) = 0

サイン, コサインで表わす

3 cos (x) - 8 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 )}

= 3 cos (x) - 8 \cdot \frac{sin ( x )}{cos ( x )}

簡素化

3 cos (x) - 8 \cdot \frac{sin ( x )}{cos ( x )} : \frac{3 cos ^{2} ( x ) - 8 sin ( x )}{cos ( x )}

3 cos (x) - 8 \cdot \frac{sin ( x )}{cos ( x )}

乗じる

8 \cdot \frac{sin ( x )}{cos ( x )} : \frac{8 sin ( x )}{cos ( x )}

8 \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 8}{cos ( x )}

= 3 cos (x) - \frac{8 sin ( 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 ( x ) \cdot 8}{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 ( x ) \cdot 8}{cos ( x )}

3 cos (x) cos (x) - sin (x) \cdot 8 = 3 cos^{2} (x) - 8 sin (x)

3 cos (x) cos (x) - sin (x) \cdot 8

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) - 8 sin (x)

= \frac{3 cos ^{2} ( x ) - 8 sin ( x )}{cos ( x )}

= \frac{3 cos ^{2} ( x ) - 8 sin ( x )}{cos ( x )}

\frac{3 cos ^{2} ( x ) - 8 sin ( x )}{cos ( x )} = 0

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

3 cos^{2} (x) - 8 sin (x) = 0

三角関数の公式を使用して書き換える

3 cos^{2} (x) - 8 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)

= 3 (1 - sin^{2} (x)) - 8 sin (x)

(1 - sin^{2} (x)) \cdot 3 - 8 sin (x) = 0

置換で解く

(1 - sin^{2} (x)) \cdot 3 - 8 sin (x) = 0

仮定:

sin (x) = u

(1 - u^{2}) \cdot 3 - 8 u = 0

(1 - u^{2}) \cdot 3 - 8 u = 0 : u = - 3, u = \frac{1}{3}

(1 - u^{2}) \cdot 3 - 8 u = 0

拡張

(1 - u^{2}) \cdot 3 - 8 u : 3 - 3 u^{2} - 8 u

(1 - u^{2}) \cdot 3 - 8 u

= 3 (1 - u^{2}) - 8 u

拡張

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

3 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

= 3 \cdot 1 - 3 u^{2}

数を乗じる:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2} - 8 u

3 - 3 u^{2} - 8 u = 0

標準的な形式で書く

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

- 3 u^{2} - 8 u + 3 = 0

解くとthe二次式

- 3 u^{2} - 8 u + 3 = 0

二次Equationの公式:

次の場合:

a = - 3, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

(- 8)^{2} - 4 (- 3) \cdot 3 = 10

(- 8)^{2} - 4 (- 3) \cdot 3

規則を適用

- (- a) = a

= (- 8)^{2} + 4 \cdot 3 \cdot 3

指数の規則を適用する:

n

が偶数であれば

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

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 3 \cdot 3

数を乗じる:

4 \cdot 3 \cdot 3 = 36

= 8^{2} + 36

8^{2} = 64

= 64 + 36

数を足す:

64 + 36 = 100

= 100

数を因数に分解する:

100 = 1 0^{2}

= 1 0^{2}

累乗根の規則を適用する:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 8 ) \pm 10}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- ( - 8 ) + 10}{2 ( - 3 )}, u_{2} = \frac{- ( - 8 ) - 10}{2 ( - 3 )}

u = \frac{- ( - 8 ) + 10}{2 ( - 3 )} : - 3

\frac{- ( - 8 ) + 10}{2 ( - 3 )}

括弧を削除する:

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

= \frac{8 + 10}{- 2 \cdot 3}

数を足す:

8 + 10 = 18

= \frac{18}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{18}{- 6}

分数の規則を適用する:

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

= - \frac{18}{6}

数を割る:

\frac{18}{6} = 3

= - 3

u = \frac{- ( - 8 ) - 10}{2 ( - 3 )} : \frac{1}{3}

\frac{- ( - 8 ) - 10}{2 ( - 3 )}

括弧を削除する:

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

= \frac{8 - 10}{- 2 \cdot 3}

数を引く:

8 - 10 = - 2

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

数を乗じる:

2 \cdot 3 = 6

= \frac{- 2}{- 6}

分数の規則を適用する:

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

= \frac{2}{6}

共通因数を約分する:

2

= \frac{1}{3}

二次equationの解:

u = - 3, u = \frac{1}{3}

代用を戻す

u = sin (x)

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

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

sin (x) = - 3 :

解なし

sin (x) = - 3

- 1 \leq sin (x) \leq 1

解なし

sin (x) = \frac{1}{3} : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

10進法形式で解を証明する

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

グラフ

人気の例

1+4sin^2(x)-5sin(x)+cos(2x)=0

1 + 4 sin^{2} (x) - 5 sin (x) + cos (2 x) = 0

cos (x) = \frac{9}{10}

cos^2(x)+4cos(x)+4=0

cos^{2} (x) + 4 cos (x) + 4 = 0

2 cos (x) = tan (x)

sin^3(a)=3sin(a)-4sin^3(a)

sin^{3} (a) = 3 sin (a) - 4 sin^{3} (a)