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 >

20sin^2(x)+27cos(x)=6

解

20 sin^{2} (x) + 27 cos (x) = 6

解

x = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn

+1

度

x = 113.57817 \dots^{\circ} + 36 0^{\circ} n, x = - 113.57817 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

20 sin^{2} (x) + 27 cos (x) = 6

両辺から

6

を引く

20 sin^{2} (x) + 27 cos (x) - 6 = 0

三角関数の公式を使用して書き換える

- 6 + 20 sin^{2} (x) + 27 cos (x)

ピタゴラスの公式を使用する:

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

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

= - 6 + 20 (1 - cos^{2} (x)) + 27 cos (x)

簡素化

- 6 + 20 (1 - cos^{2} (x)) + 27 cos (x) : 27 cos (x) - 20 cos^{2} (x) + 14

- 6 + 20 (1 - cos^{2} (x)) + 27 cos (x)

拡張

20 (1 - cos^{2} (x)) : 20 - 20 cos^{2} (x)

20 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 20, b = 1, c = cos^{2} (x)

= 20 \cdot 1 - 20 cos^{2} (x)

数を乗じる:

20 \cdot 1 = 20

= 20 - 20 cos^{2} (x)

= - 6 + 20 - 20 cos^{2} (x) + 27 cos (x)

数を足す/引く:

- 6 + 20 = 14

= 27 cos (x) - 20 cos^{2} (x) + 14

= 27 cos (x) - 20 cos^{2} (x) + 14

14 - 20 cos^{2} (x) + 27 cos (x) = 0

置換で解く

14 - 20 cos^{2} (x) + 27 cos (x) = 0

仮定:

cos (x) = u

14 - 20 u^{2} + 27 u = 0

14 - 20 u^{2} + 27 u = 0 : u = - \frac{2}{5}, u = \frac{7}{4}

14 - 20 u^{2} + 27 u = 0

標準的な形式で書く

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

- 20 u^{2} + 27 u + 14 = 0

解くとthe二次式

- 20 u^{2} + 27 u + 14 = 0

二次Equationの公式:

次の場合:

a = - 20, b = 27, c = 14

u_{1, 2} = \frac{- 27 \pm 2 7 ^{2} - 4 ( - 20 ) \cdot 14}{2 ( - 20 )}

u_{1, 2} = \frac{- 27 \pm 2 7 ^{2} - 4 ( - 20 ) \cdot 14}{2 ( - 20 )}

2 7^{2} - 4 (- 20) \cdot 14 = 43

2 7^{2} - 4 (- 20) \cdot 14

規則を適用

- (- a) = a

= 2 7^{2} + 4 \cdot 20 \cdot 14

数を乗じる:

4 \cdot 20 \cdot 14 = 1120

= 2 7^{2} + 1120

2 7^{2} = 729

= 729 + 1120

数を足す:

729 + 1120 = 1849

= 1849

数を因数に分解する:

1849 = 4 3^{2}

= 4 3^{2}

累乗根の規則を適用する:

n a^{n} = a

4 3^{2} = 43

= 43

u_{1, 2} = \frac{- 27 \pm 43}{2 ( - 20 )}

解を分離する

u_{1} = \frac{- 27 + 43}{2 ( - 20 )}, u_{2} = \frac{- 27 - 43}{2 ( - 20 )}

u = \frac{- 27 + 43}{2 ( - 20 )} : - \frac{2}{5}

\frac{- 27 + 43}{2 ( - 20 )}

括弧を削除する:

(- a) = - a

= \frac{- 27 + 43}{- 2 \cdot 20}

数を足す/引く:

- 27 + 43 = 16

= \frac{16}{- 2 \cdot 20}

数を乗じる:

2 \cdot 20 = 40

= \frac{16}{- 40}

分数の規則を適用する:

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

= - \frac{16}{40}

共通因数を約分する:

8

= - \frac{2}{5}

u = \frac{- 27 - 43}{2 ( - 20 )} : \frac{7}{4}

\frac{- 27 - 43}{2 ( - 20 )}

括弧を削除する:

(- a) = - a

= \frac{- 27 - 43}{- 2 \cdot 20}

数を引く:

- 27 - 43 = - 70

= \frac{- 70}{- 2 \cdot 20}

数を乗じる:

2 \cdot 20 = 40

= \frac{- 70}{- 40}

分数の規則を適用する:

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

= \frac{70}{40}

共通因数を約分する:

10

= \frac{7}{4}

二次equationの解:

u = - \frac{2}{5}, u = \frac{7}{4}

代用を戻す

u = cos (x)

cos (x) = - \frac{2}{5}, cos (x) = \frac{7}{4}

cos (x) = - \frac{2}{5}, cos (x) = \frac{7}{4}

cos (x) = - \frac{2}{5} : x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

cos (x) = - \frac{2}{5}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{2}{5}

以下の一般解

cos (x) = - \frac{2}{5}

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

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

cos (x) = \frac{7}{4} :

解なし

cos (x) = \frac{7}{4}

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

x = arccos (- \frac{2}{5}) + 2 πn, x = - arccos (- \frac{2}{5}) + 2 πn

10進法形式で解を証明する

x = 1.98231 \dots + 2 πn, x = - 1.98231 \dots + 2 πn

グラフ

人気の例

-8csc^2(x)-4cot(x)=-12

- 8 csc^{2} (x) - 4 cot (x) = - 12

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

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

sin (θ) = \frac{5}{3}

14sin^2(x)+5sin(x)-1=0

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

sin(x)= 3/5 sin(x+pi/2)+cos(pi-x)

sin (x) = \frac{3}{5} sin (x + \frac{π}{2}) + cos (π - x)