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人気のある三角関数 >

3sin^2(x)=cos^2(x)-4cos(x)

解

3 sin^{2} (x) = cos^{2} (x) - 4 cos (x)

解

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

+1

度

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin^{2} (x) = cos^{2} (x) - 4 cos (x)

両辺から

cos^{2} (x) - 4 cos (x)

を引く

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

三角関数の公式を使用して書き換える

- cos^{2} (x) + 3 sin^{2} (x) + 4 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)

= - cos^{2} (x) + 3 (1 - cos^{2} (x)) + 4 cos (x)

簡素化

- cos^{2} (x) + 3 (1 - cos^{2} (x)) + 4 cos (x) : - 4 cos^{2} (x) + 4 cos (x) + 3

- cos^{2} (x) + 3 (1 - cos^{2} (x)) + 4 cos (x)

拡張

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

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

数を乗じる:

3 \cdot 1 = 3

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

= - cos^{2} (x) + 3 - 3 cos^{2} (x) + 4 cos (x)

簡素化

- cos^{2} (x) + 3 - 3 cos^{2} (x) + 4 cos (x) : - 4 cos^{2} (x) + 4 cos (x) + 3

- cos^{2} (x) + 3 - 3 cos^{2} (x) + 4 cos (x)

条件のようなグループ

= - cos^{2} (x) - 3 cos^{2} (x) + 4 cos (x) + 3

類似した元を足す:

- cos^{2} (x) - 3 cos^{2} (x) = - 4 cos^{2} (x)

= - 4 cos^{2} (x) + 4 cos (x) + 3

= - 4 cos^{2} (x) + 4 cos (x) + 3

= - 4 cos^{2} (x) + 4 cos (x) + 3

3 + 4 cos (x) - 4 cos^{2} (x) = 0

置換で解く

3 + 4 cos (x) - 4 cos^{2} (x) = 0

仮定:

cos (x) = u

3 + 4 u - 4 u^{2} = 0

3 + 4 u - 4 u^{2} = 0 : u = - \frac{1}{2}, u = \frac{3}{2}

3 + 4 u - 4 u^{2} = 0

標準的な形式で書く

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

- 4 u^{2} + 4 u + 3 = 0

解くとthe二次式

- 4 u^{2} + 4 u + 3 = 0

二次Equationの公式:

次の場合:

a = - 4, b = 4, c = 3

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

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

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

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

規則を適用

- (- a) = a

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

数を乗じる:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数を足す:

16 + 48 = 64

= 64

数を因数に分解する:

64 = 8^{2}

= 8^{2}

累乗根の規則を適用する:

n a^{n} = a

8^{2} = 8

= 8

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

解を分離する

u_{1} = \frac{- 4 + 8}{2 ( - 4 )}, u_{2} = \frac{- 4 - 8}{2 ( - 4 )}

u = \frac{- 4 + 8}{2 ( - 4 )} : - \frac{1}{2}

\frac{- 4 + 8}{2 ( - 4 )}

括弧を削除する:

(- a) = - a

= \frac{- 4 + 8}{- 2 \cdot 4}

数を足す/引く:

- 4 + 8 = 4

= \frac{4}{- 2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{4}{- 8}

分数の規則を適用する:

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

= - \frac{4}{8}

共通因数を約分する:

4

= - \frac{1}{2}

u = \frac{- 4 - 8}{2 ( - 4 )} : \frac{3}{2}

\frac{- 4 - 8}{2 ( - 4 )}

括弧を削除する:

(- a) = - a

= \frac{- 4 - 8}{- 2 \cdot 4}

数を引く:

- 4 - 8 = - 12

= \frac{- 12}{- 2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{- 12}{- 8}

分数の規則を適用する:

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

= \frac{12}{8}

共通因数を約分する:

4

= \frac{3}{2}

二次equationの解:

u = - \frac{1}{2}, u = \frac{3}{2}

代用を戻す

u = cos (x)

cos (x) = - \frac{1}{2}, cos (x) = \frac{3}{2}

cos (x) = - \frac{1}{2}, cos (x) = \frac{3}{2}

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

以下の一般解

cos (x) = - \frac{1}{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}

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

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

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

解なし

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

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

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

グラフ

人気の例

cos (x) = 0.9224

-9sin^2(θ)+3cos(θ)-2=-9

- 9 sin^{2} (θ) + 3 cos (θ) - 2 = - 9

4sin(x)=4cos(x)

4 sin (x) = 4 cos (x)

6 sin (3 x) = 0

tan (x) = \frac{7}{24}