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人気のある三角関数 >

5sin(x)-3cos(2x)+3=0

解

5 sin (x) - 3 cos (2 x) + 3 = 0

解

x = 2 πn, x = π + 2 πn, x = - 0.98511 \dots + 2 πn, x = π + 0.98511 \dots + 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = - 56.44269 \dots^{\circ} + 36 0^{\circ} n, x = 236.44269 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

5 sin (x) - 3 cos (2 x) + 3 = 0

三角関数の公式を使用して書き換える

3 - 3 cos (2 x) + 5 sin (x)

2倍角の公式を使用:

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

= 3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x)

簡素化

3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x) : 6 sin^{2} (x) + 5 sin (x)

3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x)

拡張

- 3 (1 - 2 sin^{2} (x)) : - 3 + 6 sin^{2} (x)

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

分配法則を適用する:

a (b - c) = ab - a c

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

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

マイナス・プラスの規則を適用する

- (- a) = a

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

簡素化

- 3 \cdot 1 + 3 \cdot 2 sin^{2} (x) : - 3 + 6 sin^{2} (x)

- 3 \cdot 1 + 3 \cdot 2 sin^{2} (x)

数を乗じる:

3 \cdot 1 = 3

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

数を乗じる:

3 \cdot 2 = 6

= - 3 + 6 sin^{2} (x)

= - 3 + 6 sin^{2} (x)

= 3 - 3 + 6 sin^{2} (x) + 5 sin (x)

3 - 3 = 0

= 6 sin^{2} (x) + 5 sin (x)

= 6 sin^{2} (x) + 5 sin (x)

5 sin (x) + 6 sin^{2} (x) = 0

置換で解く

5 sin (x) + 6 sin^{2} (x) = 0

仮定:

sin (x) = u

5 u + 6 u^{2} = 0

5 u + 6 u^{2} = 0 : u = 0, u = - \frac{5}{6}

5 u + 6 u^{2} = 0

標準的な形式で書く

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

6 u^{2} + 5 u = 0

解くとthe二次式

6 u^{2} + 5 u = 0

二次Equationの公式:

次の場合:

a = 6, b = 5, c = 0

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 6 \cdot 0}{2 \cdot 6}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 6 \cdot 0}{2 \cdot 6}

5^{2} - 4 \cdot 6 \cdot 0 = 5

5^{2} - 4 \cdot 6 \cdot 0

規則を適用

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

累乗根の規則を適用する:

n a^{n} = a,

, 以下を想定

a \geq 0

= 5

u_{1, 2} = \frac{- 5 \pm 5}{2 \cdot 6}

解を分離する

u_{1} = \frac{- 5 + 5}{2 \cdot 6}, u_{2} = \frac{- 5 - 5}{2 \cdot 6}

u = \frac{- 5 + 5}{2 \cdot 6} : 0

\frac{- 5 + 5}{2 \cdot 6}

数を足す/引く:

- 5 + 5 = 0

= \frac{0}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{0}{12}

規則を適用

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

= 0

u = \frac{- 5 - 5}{2 \cdot 6} : - \frac{5}{6}

\frac{- 5 - 5}{2 \cdot 6}

数を引く:

- 5 - 5 = - 10

= \frac{- 10}{2 \cdot 6}

数を乗じる:

2 \cdot 6 = 12

= \frac{- 10}{12}

分数の規則を適用する:

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

= - \frac{10}{12}

共通因数を約分する:

2

= - \frac{5}{6}

二次equationの解:

u = 0, u = - \frac{5}{6}

代用を戻す

u = sin (x)

sin (x) = 0, sin (x) = - \frac{5}{6}

sin (x) = 0, sin (x) = - \frac{5}{6}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

以下の一般解

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = - \frac{5}{6} : x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

sin (x) = - \frac{5}{6}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{5}{6}

以下の一般解

sin (x) = - \frac{5}{6}

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

x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn, x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

10進法形式で解を証明する

x = 2 πn, x = π + 2 πn, x = - 0.98511 \dots + 2 πn, x = π + 0.98511 \dots + 2 πn

グラフ

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