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人気のある三角関数 >

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

解

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

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

両辺から

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

を引く

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

三角関数の公式を使用して書き換える

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

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

簡素化

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

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

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

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

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

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

簡素化

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

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

類似した元を足す:

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

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

数を引く:

2 - 1 = 1

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

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

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

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

置換で解く

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

仮定:

cos (x) = u

1 + 4 u + 4 u^{2} = 0

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

1 + 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 + 1 = 0

解くとthe二次式

4 u^{2} + 4 u + 1 = 0

二次Equationの公式:

次の場合:

a = 4, b = 4, c = 1

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

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

4^{2} - 4 \cdot 4 \cdot 1 = 0

4^{2} - 4 \cdot 4 \cdot 1

数を乗じる:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

数を引く:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 \cdot 4}

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

\frac{- 4}{2 \cdot 4} = - \frac{1}{2}

\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{1}{2}

二次equationの解:

u = - \frac{1}{2}

代用を戻す

u = cos (x)

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

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

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

50sin(x)+15cos(x)=40,0<x<pi

50 sin (x) + 15 cos (x) = 40, 0 < x < π

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10 sin (x) cos (x) = 6 cos (x)

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sin (x) + cos (x) = sec (x)

sin (x) = - 0.2761

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