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人気のある三角関数 >

48sin^2(x)=48-24cos(x)

解

48 sin^{2} (x) = 48 - 24 cos (x)

解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

解答ステップ

48 sin^{2} (x) = 48 - 24 cos (x)

両辺から

48 - 24 cos (x)

を引く

48 sin^{2} (x) - 48 + 24 cos (x) = 0

三角関数の公式を使用して書き換える

- 48 + 24 cos (x) + 48 sin^{2} (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)

= - 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

簡素化

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x)) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 (1 - cos^{2} (x))

拡張

48 (1 - cos^{2} (x)) : 48 - 48 cos^{2} (x)

48 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 48, b = 1, c = cos^{2} (x)

= 48 \cdot 1 - 48 cos^{2} (x)

数を乗じる:

48 \cdot 1 = 48

= 48 - 48 cos^{2} (x)

= - 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

簡素化

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x) : 24 cos (x) - 48 cos^{2} (x)

- 48 + 24 cos (x) + 48 - 48 cos^{2} (x)

条件のようなグループ

= 24 cos (x) - 48 cos^{2} (x) - 48 + 48

- 48 + 48 = 0

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

= 24 cos (x) - 48 cos^{2} (x)

24 cos (x) - 48 cos^{2} (x) = 0

置換で解く

24 cos (x) - 48 cos^{2} (x) = 0

仮定:

cos (x) = u

24 u - 48 u^{2} = 0

24 u - 48 u^{2} = 0 : u = 0, u = \frac{1}{2}

24 u - 48 u^{2} = 0

標準的な形式で書く

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

- 48 u^{2} + 24 u = 0

解くとthe二次式

- 48 u^{2} + 24 u = 0

二次Equationの公式:

次の場合:

a = - 48, b = 24, c = 0

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 ( - 48 ) \cdot 0}{2 ( - 48 )}

2 4^{2} - 4 (- 48) \cdot 0 = 24

2 4^{2} - 4 (- 48) \cdot 0

規則を適用

- (- a) = a

= 2 4^{2} + 4 \cdot 48 \cdot 0

規則を適用

0 \cdot a = 0

= 2 4^{2} + 0

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

= 2 4^{2}

累乗根の規則を適用する:

n a^{n} = a,

, 以下を想定

a \geq 0

= 24

u_{1, 2} = \frac{- 24 \pm 24}{2 ( - 48 )}

解を分離する

u_{1} = \frac{- 24 + 24}{2 ( - 48 )}, u_{2} = \frac{- 24 - 24}{2 ( - 48 )}

u = \frac{- 24 + 24}{2 ( - 48 )} : 0

\frac{- 24 + 24}{2 ( - 48 )}

括弧を削除する:

(- a) = - a

= \frac{- 24 + 24}{- 2 \cdot 48}

数を足す/引く:

- 24 + 24 = 0

= \frac{0}{- 2 \cdot 48}

数を乗じる:

2 \cdot 48 = 96

= \frac{0}{- 96}

分数の規則を適用する:

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

= - \frac{0}{96}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 24 - 24}{2 ( - 48 )} : \frac{1}{2}

\frac{- 24 - 24}{2 ( - 48 )}

括弧を削除する:

(- a) = - a

= \frac{- 24 - 24}{- 2 \cdot 48}

数を引く:

- 24 - 24 = - 48

= \frac{- 48}{- 2 \cdot 48}

数を乗じる:

2 \cdot 48 = 96

= \frac{- 48}{- 96}

分数の規則を適用する:

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

= \frac{48}{96}

共通因数を約分する:

48

= \frac{1}{2}

二次equationの解:

u = 0, u = \frac{1}{2}

代用を戻す

u = cos (x)

cos (x) = 0, cos (x) = \frac{1}{2}

cos (x) = 0, cos (x) = \frac{1}{2}

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

以下の一般解

cos (x) = 0

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

すべての解を組み合わせる

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

グラフ

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