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人気のある三角関数 >

8sin^2(2x)-2sin(2x)-1=0

解

8 \cdot sin^{2} (2 x) - 2 \cdot sin (2 x) - 1 = 0

解

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{0.25268 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.25268 \dots}{2} + πn

+1

度

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n, x = - 7.23875 \dots^{\circ} + 18 0^{\circ} n, x = 97.23875 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

8 sin^{2} (2 x) - 2 sin (2 x) - 1 = 0

置換で解く

8 sin^{2} (2 x) - 2 sin (2 x) - 1 = 0

仮定:

sin (2 x) = u

8 u^{2} - 2 u - 1 = 0

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

8 u^{2} - 2 u - 1 = 0

解くとthe二次式

8 u^{2} - 2 u - 1 = 0

二次Equationの公式:

次の場合:

a = 8, b = - 2, c = - 1

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

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

(- 2)^{2} - 4 \cdot 8 (- 1) = 6

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

規則を適用

- (- a) = a

= (- 2)^{2} + 4 \cdot 8 \cdot 1

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 8 \cdot 1

数を乗じる:

4 \cdot 8 \cdot 1 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数を足す:

4 + 32 = 36

= 36

数を因数に分解する:

36 = 6^{2}

= 6^{2}

累乗根の規則を適用する:

n a^{n} = a

6^{2} = 6

= 6

u_{1, 2} = \frac{- ( - 2 ) \pm 6}{2 \cdot 8}

解を分離する

u_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 8}, u_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 8}

u = \frac{- ( - 2 ) + 6}{2 \cdot 8} : \frac{1}{2}

\frac{- ( - 2 ) + 6}{2 \cdot 8}

規則を適用

- (- a) = a

= \frac{2 + 6}{2 \cdot 8}

数を足す:

2 + 6 = 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{8}{16}

共通因数を約分する:

8

= \frac{1}{2}

u = \frac{- ( - 2 ) - 6}{2 \cdot 8} : - \frac{1}{4}

\frac{- ( - 2 ) - 6}{2 \cdot 8}

規則を適用

- (- a) = a

= \frac{2 - 6}{2 \cdot 8}

数を引く:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 8}

数を乗じる:

2 \cdot 8 = 16

= \frac{- 4}{16}

分数の規則を適用する:

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

= - \frac{4}{16}

共通因数を約分する:

4

= - \frac{1}{4}

二次equationの解:

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

代用を戻す

u = sin (2 x)

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

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

sin (2 x) = \frac{1}{2} : x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

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

以下の一般解

sin (2 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}

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

解く

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

解く

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

2 x = \frac{5 π}{6} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{6 \cdot 2}

数を乗じる:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

sin (2 x) = - \frac{1}{4} : x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

2 x = arcsin (- \frac{1}{4}) + 2 πn, 2 x = π + arcsin (\frac{1}{4}) + 2 πn

2 x = arcsin (- \frac{1}{4}) + 2 πn, 2 x = π + arcsin (\frac{1}{4}) + 2 πn

解く

2 x = arcsin (- \frac{1}{4}) + 2 πn : x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn

2 x = arcsin (- \frac{1}{4}) + 2 πn

簡素化

arcsin (- \frac{1}{4}) + 2 πn : - arcsin (\frac{1}{4}) + 2 πn

arcsin (- \frac{1}{4}) + 2 πn

次のプロパティを使用する:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{4}) = - arcsin (\frac{1}{4})

= - arcsin (\frac{1}{4}) + 2 πn

2 x = - arcsin (\frac{1}{4}) + 2 πn

以下で両辺を割る

2

2 x = - arcsin (\frac{1}{4}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

簡素化

x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn

解く

2 x = π + arcsin (\frac{1}{4}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

2 x = π + arcsin (\frac{1}{4}) + 2 πn

以下で両辺を割る

2

2 x = π + arcsin (\frac{1}{4}) + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

簡素化

x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

すべての解を組み合わせる

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{arcsin ( \frac{1}{4} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{1}{4} )}{2} + πn

10進法形式で解を証明する

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn, x = - \frac{0.25268 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.25268 \dots}{2} + πn

グラフ

人気の例

4 - 2 cos (2 x) = 0

solvefor x,2cos^2(x)+sin(x)=1

so l v e f or x, 2 cos^{2} (x) + sin (x) = 1

2cos^2(x)+6cos(x)=-1

2 cos^{2} (x) + 6 cos (x) = - 1

8sin^2(x)-2cos(x)=5

8 sin^{2} (x) - 2 cos (x) = 5

22=18+8cos((x+9)/(12)pi)

22 = 18 + 8 cos (\frac{x + 9}{12} π)