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人気のある三角関数 >

solvefor x,4sin^4(x)-8sin^2(x)+3=0

解

解く

x, 4 sin^{4} (x) - 8 sin^{2} (x) + 3 = 0

解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

解答ステップ

4 sin^{4} (x) - 8 sin^{2} (x) + 3 = 0

置換で解く

4 sin^{4} (x) - 8 sin^{2} (x) + 3 = 0

仮定:

sin (x) = u

4 u^{4} - 8 u^{2} + 3 = 0

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

4 u^{4} - 8 u^{2} + 3 = 0

equationを

v = u^{2}

と以下で書き換える:

v^{2} = u^{4}

4 v^{2} - 8 v + 3 = 0

解く

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

4 v^{2} - 8 v + 3 = 0

解くとthe二次式

4 v^{2} - 8 v + 3 = 0

二次Equationの公式:

次の場合:

a = 4, b = - 8, c = 3

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

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

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

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

指数の規則を適用する:

n

が偶数であれば

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

(- 8)^{2} = 8^{2}

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

数を乗じる:

4 \cdot 4 \cdot 3 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

数を引く:

64 - 48 = 16

= 16

数を因数に分解する:

16 = 4^{2}

= 4^{2}

累乗根の規則を適用する:

n a^{n} = a

4^{2} = 4

= 4

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

解を分離する

v_{1} = \frac{- ( - 8 ) + 4}{2 \cdot 4}, v_{2} = \frac{- ( - 8 ) - 4}{2 \cdot 4}

v = \frac{- ( - 8 ) + 4}{2 \cdot 4} : \frac{3}{2}

\frac{- ( - 8 ) + 4}{2 \cdot 4}

規則を適用

- (- a) = a

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

数を足す:

8 + 4 = 12

= \frac{12}{2 \cdot 4}

数を乗じる:

2 \cdot 4 = 8

= \frac{12}{8}

共通因数を約分する:

4

= \frac{3}{2}

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

\frac{- ( - 8 ) - 4}{2 \cdot 4}

規則を適用

- (- a) = a

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

数を引く:

8 - 4 = 4

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

数を乗じる:

2 \cdot 4 = 8

= \frac{4}{8}

共通因数を約分する:

4

= \frac{1}{2}

二次equationの解:

v = \frac{3}{2}, v = \frac{1}{2}

v = \frac{3}{2}, v = \frac{1}{2}

再び

v = u^{2}

に置き換えて以下を解く:

u

解く

u^{2} = \frac{3}{2} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{2}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

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

解く

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

u^{2} = \frac{1}{2}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

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

解答は

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

代用を戻す

u = sin (x)

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

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

sin (x) = \frac{3}{2} :

解なし

sin (x) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

解なし

sin (x) = - \frac{3}{2} :

解なし

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

- 1 \leq sin (x) \leq 1

解なし

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 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{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

すべての解を組み合わせる

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

グラフ

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