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人気のある三角関数 >

4cos^2(x/2)-3=0

解

4 cos^{2} (\frac{x}{2}) - 3 = 0

解

x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

+1

度

x = 6 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n, x = 30 0^{\circ} + 72 0^{\circ} n, x = 42 0^{\circ} + 72 0^{\circ} n

解答ステップ

4 cos^{2} (\frac{x}{2}) - 3 = 0

置換で解く

4 cos^{2} (\frac{x}{2}) - 3 = 0

仮定:

cos (\frac{x}{2}) = u

4 u^{2} - 3 = 0

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

4 u^{2} - 3 = 0

3

を右側に移動します

4 u^{2} - 3 = 0

両辺に

3

を足す

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

簡素化

4 u^{2} = 3

4 u^{2} = 3

以下で両辺を割る

4

4 u^{2} = 3

以下で両辺を割る

4

\frac{4 u ^{2}}{4} = \frac{3}{4}

簡素化

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

x^{2} = f (a)

の場合, 解は

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

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

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

簡素化

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

代用を戻す

u = cos (\frac{x}{2})

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

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

cos (\frac{x}{2}) = \frac{3}{2} : x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn

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

以下の一般解

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

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

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

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

解く

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{π}{6} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

分数を乗じる:

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

= \frac{π 2}{6}

共通因数を約分する:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

解く

\frac{x}{2} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{x}{2} = \frac{11 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{11 π}{6} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

分数を乗じる:

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

= \frac{11 π 2}{6}

数を乗じる:

11 \cdot 2 = 22

= \frac{22 π}{6}

共通因数を約分する:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn

cos (\frac{x}{2}) = - \frac{3}{2} : x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

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

以下の一般解

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

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

\frac{x}{2} = \frac{5 π}{6} + 2 πn, \frac{x}{2} = \frac{7 π}{6} + 2 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn, \frac{x}{2} = \frac{7 π}{6} + 2 πn

解く

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

分数を乗じる:

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

= \frac{5 π 2}{6}

数を乗じる:

5 \cdot 2 = 10

= \frac{10 π}{6}

共通因数を約分する:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

解く

\frac{x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{x}{2} = \frac{7 π}{6} + 2 πn

以下で両辺を乗じる:

2

\frac{x}{2} = \frac{7 π}{6} + 2 πn

以下で両辺を乗じる:

2

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

簡素化

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

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

分数を乗じる:

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

= \frac{7 π 2}{6}

数を乗じる:

7 \cdot 2 = 14

= \frac{14 π}{6}

共通因数を約分する:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数を乗じる:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

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

すべての解を組み合わせる

x = \frac{π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn, x = \frac{7 π}{3} + 4 πn

グラフ

人気の例

sin(2θ)+cos(θ)=0,0<= θ<= 2pi

sin (2 θ) + cos (θ) = 0, 0 \leq θ \leq 2 π

sin(10)=2sin(a)cos(a)

sin (1 0^{\circ}) = 2 sin (a) cos (a)

cos^2(x)=2+2sin(x),0<= x<= 2pi

cos^{2} (x) = 2 + 2 sin (x), 0 \leq x \leq 2 π

cos^{2} (x) = \frac{3}{5}

1-sec^2(x)=tan^2(x)

1 - sec^{2} (x) = tan^{2} (x)