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人気のある三角関数 >

4cos(2x)+2sin(x)=1,0<= x<= 360

解

4 cos (2 x) + 2 sin (x) = 1, 0^{\circ} \leq x \leq 36 0^{\circ}

解

x = 21 0^{\circ}, x = 33 0^{\circ}, x = 0.84806 \dots, x = 18 0^{\circ} - 0.84806 \dots

+1

ラジアン

x = \frac{7 π}{6}, x = \frac{11 π}{6}, x = 0.84806 \dots, x = π - 0.84806 \dots

解答ステップ

4 cos (2 x) + 2 sin (x) = 1, 0^{\circ} \leq x \leq 36 0^{\circ}

両辺から

1

を引く

4 cos (2 x) + 2 sin (x) - 1 = 0

三角関数の公式を使用して書き換える

- 1 + 2 sin (x) + 4 cos (2 x)

2倍角の公式を使用:

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

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

簡素化

- 1 + 2 sin (x) + 4 (1 - 2 sin^{2} (x)) : 2 sin (x) - 8 sin^{2} (x) + 3

- 1 + 2 sin (x) + 4 (1 - 2 sin^{2} (x))

拡張

4 (1 - 2 sin^{2} (x)) : 4 - 8 sin^{2} (x)

4 (1 - 2 sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 4, b = 1, c = 2 sin^{2} (x)

= 4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

簡素化

4 \cdot 1 - 4 \cdot 2 sin^{2} (x) : 4 - 8 sin^{2} (x)

4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

数を乗じる:

4 \cdot 1 = 4

= 4 - 4 \cdot 2 sin^{2} (x)

数を乗じる:

4 \cdot 2 = 8

= 4 - 8 sin^{2} (x)

= 4 - 8 sin^{2} (x)

= - 1 + 2 sin (x) + 4 - 8 sin^{2} (x)

簡素化

- 1 + 2 sin (x) + 4 - 8 sin^{2} (x) : 2 sin (x) - 8 sin^{2} (x) + 3

- 1 + 2 sin (x) + 4 - 8 sin^{2} (x)

条件のようなグループ

= 2 sin (x) - 8 sin^{2} (x) - 1 + 4

数を足す/引く:

- 1 + 4 = 3

= 2 sin (x) - 8 sin^{2} (x) + 3

= 2 sin (x) - 8 sin^{2} (x) + 3

= 2 sin (x) - 8 sin^{2} (x) + 3

3 + 2 sin (x) - 8 sin^{2} (x) = 0

置換で解く

3 + 2 sin (x) - 8 sin^{2} (x) = 0

仮定:

sin (x) = u

3 + 2 u - 8 u^{2} = 0

3 + 2 u - 8 u^{2} = 0 : u = - \frac{1}{2}, u = \frac{3}{4}

3 + 2 u - 8 u^{2} = 0

標準的な形式で書く

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

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = - 8, b = 2, c = 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 3}{2 ( - 8 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 3}{2 ( - 8 )}

2^{2} - 4 (- 8) \cdot 3 = 10

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

規則を適用

- (- a) = a

= 2^{2} + 4 \cdot 8 \cdot 3

数を乗じる:

4 \cdot 8 \cdot 3 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

数を足す:

4 + 96 = 100

= 100

数を因数に分解する:

100 = 1 0^{2}

= 1 0^{2}

累乗根の規則を適用する:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 ( - 8 )}

解を分離する

u_{1} = \frac{- 2 + 10}{2 ( - 8 )}, u_{2} = \frac{- 2 - 10}{2 ( - 8 )}

u = \frac{- 2 + 10}{2 ( - 8 )} : - \frac{1}{2}

\frac{- 2 + 10}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

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

数を足す/引く:

- 2 + 10 = 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{- 16}

分数の規則を適用する:

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

= - \frac{8}{16}

共通因数を約分する:

8

= - \frac{1}{2}

u = \frac{- 2 - 10}{2 ( - 8 )} : \frac{3}{4}

\frac{- 2 - 10}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

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

数を引く:

- 2 - 10 = - 12

= \frac{- 12}{- 2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 12}{- 16}

分数の規則を適用する:

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

= \frac{12}{16}

共通因数を約分する:

4

= \frac{3}{4}

二次equationの解:

u = - \frac{1}{2}, u = \frac{3}{4}

代用を戻す

u = sin (x)

sin (x) = - \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{2}, 0 \leq x \leq 36 0^{\circ} : x = 21 0^{\circ}, x = 33 0^{\circ}

sin (x) = - \frac{1}{2}, 0 \leq x \leq 36 0^{\circ}

以下の一般解

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

sin (x)

36 0^{\circ} n

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

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

範囲の解答

0 \leq x \leq 36 0^{\circ}

x = 21 0^{\circ}, x = 33 0^{\circ}

sin (x) = \frac{3}{4}, 0 \leq x \leq 36 0^{\circ} : x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

sin (x) = \frac{3}{4}, 0 \leq x \leq 36 0^{\circ}

三角関数の逆数プロパティを適用する

sin (x) = \frac{3}{4}

以下の一般解

sin (x) = \frac{3}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

x = arcsin (\frac{3}{4}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{3}{4}) + 36 0^{\circ} n

範囲の解答

0 \leq x \leq 36 0^{\circ}

x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

すべての解を組み合わせる

x = 21 0^{\circ}, x = 33 0^{\circ}, x = arcsin (\frac{3}{4}), x = 18 0^{\circ} - arcsin (\frac{3}{4})

10進法形式で解を証明する

x = 21 0^{\circ}, x = 33 0^{\circ}, x = 0.84806 \dots, x = 18 0^{\circ} - 0.84806 \dots

グラフ

人気の例

3 csc (θ) + 5 = 0

cos(x/3)-cos(x/(15))=0

cos (\frac{x}{3}) - cos (\frac{x}{15}) = 0

arctan (x) = 3

tan(x)cos(x)=cos(x)

tan (x) cos (x) = cos (x)

4cos^2(θ)-2=-1

4 cos^{2} (θ) - 2 = - 1