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人気のある三角関数 >

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

解

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

解

x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度

x = 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 156.42182 \dots^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

5 cos (2 x)

を引く

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

三角関数の公式を使用して書き換える

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

2倍角の公式を使用:

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

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

簡素化

3 + sin (x) - 5 (1 - 2 sin^{2} (x)) : 10 sin^{2} (x) + sin (x) - 2

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

拡張

- 5 (1 - 2 sin^{2} (x)) : - 5 + 10 sin^{2} (x)

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

分配法則を適用する:

a (b - c) = ab - a c

a = - 5, b = 1, c = 2 sin^{2} (x)

= - 5 \cdot 1 - (- 5) \cdot 2 sin^{2} (x)

マイナス・プラスの規則を適用する

- (- a) = a

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

簡素化

- 5 \cdot 1 + 5 \cdot 2 sin^{2} (x) : - 5 + 10 sin^{2} (x)

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

数を乗じる:

5 \cdot 1 = 5

= - 5 + 5 \cdot 2 sin^{2} (x)

数を乗じる:

5 \cdot 2 = 10

= - 5 + 10 sin^{2} (x)

= - 5 + 10 sin^{2} (x)

= 3 + sin (x) - 5 + 10 sin^{2} (x)

簡素化

3 + sin (x) - 5 + 10 sin^{2} (x) : 10 sin^{2} (x) + sin (x) - 2

3 + sin (x) - 5 + 10 sin^{2} (x)

条件のようなグループ

= sin (x) + 10 sin^{2} (x) + 3 - 5

数を足す/引く:

3 - 5 = - 2

= 10 sin^{2} (x) + sin (x) - 2

= 10 sin^{2} (x) + sin (x) - 2

= 10 sin^{2} (x) + sin (x) - 2

- 2 + sin (x) + 10 sin^{2} (x) = 0

置換で解く

- 2 + sin (x) + 10 sin^{2} (x) = 0

仮定:

sin (x) = u

- 2 + u + 10 u^{2} = 0

- 2 + u + 10 u^{2} = 0 : u = \frac{2}{5}, u = - \frac{1}{2}

- 2 + u + 10 u^{2} = 0

標準的な形式で書く

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

10 u^{2} + u - 2 = 0

解くとthe二次式

10 u^{2} + u - 2 = 0

二次Equationの公式:

次の場合:

a = 10, b = 1, c = - 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 10 ( - 2 )}{2 \cdot 10}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 10 ( - 2 )}{2 \cdot 10}

1^{2} - 4 \cdot 10 (- 2) = 9

1^{2} - 4 \cdot 10 (- 2)

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 10 (- 2)

規則を適用

- (- a) = a

= 1 + 4 \cdot 10 \cdot 2

数を乗じる:

4 \cdot 10 \cdot 2 = 80

= 1 + 80

数を足す:

1 + 80 = 81

= 81

数を因数に分解する:

81 = 9^{2}

= 9^{2}

累乗根の規則を適用する:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 1 \pm 9}{2 \cdot 10}

解を分離する

u_{1} = \frac{- 1 + 9}{2 \cdot 10}, u_{2} = \frac{- 1 - 9}{2 \cdot 10}

u = \frac{- 1 + 9}{2 \cdot 10} : \frac{2}{5}

\frac{- 1 + 9}{2 \cdot 10}

数を足す/引く:

- 1 + 9 = 8

= \frac{8}{2 \cdot 10}

数を乗じる:

2 \cdot 10 = 20

= \frac{8}{20}

共通因数を約分する:

4

= \frac{2}{5}

u = \frac{- 1 - 9}{2 \cdot 10} : - \frac{1}{2}

\frac{- 1 - 9}{2 \cdot 10}

数を引く:

- 1 - 9 = - 10

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

数を乗じる:

2 \cdot 10 = 20

= \frac{- 10}{20}

分数の規則を適用する:

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

= - \frac{10}{20}

共通因数を約分する:

10

= - \frac{1}{2}

二次equationの解:

u = \frac{2}{5}, u = - \frac{1}{2}

代用を戻す

u = sin (x)

sin (x) = \frac{2}{5}, sin (x) = - \frac{1}{2}

sin (x) = \frac{2}{5}, sin (x) = - \frac{1}{2}

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

sin (x) = \frac{2}{5}

三角関数の逆数プロパティを適用する

sin (x) = \frac{2}{5}

以下の一般解

sin (x) = \frac{2}{5}

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

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

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

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

以下の一般解

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

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 = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

10進法形式で解を証明する

x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

グラフ

人気の例

cos (x) = \frac{15}{20}

asin(x)+bcos(x)=0

a sin (x) + b cos (x) = 0

tan (2 x) = - 0.025

4 7^{\circ} = cos (x), x

sin(θ)=-(sqrt(5))/3 cos(2θ)

sin (θ) = - \frac{5}{3} cos (2 θ)