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受欢迎的三角函数 >

2cos^2(x)-sqrt(3cos(x))=0

解答

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

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.43097 \dots + 2 πn, x = 2 π - 0.43097 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 24.69285 \dots^{\circ} + 36 0^{\circ} n, x = 335.30714 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

2 u^{2} - 3 u = 0

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

2 u^{2} - 3 u = 0

去除平方根

2 u^{2} - 3 u = 0

两边减去

2 u^{2}

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

化简

- 3 u = - 2 u^{2}

两边进行平方:

3 u = 4 u^{4}

2 u^{2} - 3 u = 0

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

展开

(- 3 u)^{2} : 3 u

(- 3 u)^{2}

使用指数法则:

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

若

n

是偶数

(- 3 u)^{2} = (3 u)^{2}

= (3 u)^{2}

使用根式运算法则:

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

= ((3 u)^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (3 u)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3 u

展开

(- 2 u^{2})^{2} : 4 u^{4}

(- 2 u^{2})^{2}

使用指数法则:

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

若

n

是偶数

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

= (2 u^{2})^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (u^{2})^{2}

(u^{2})^{2} : u^{4}

使用指数法则:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= u^{4}

= 2^{2} u^{4}

2^{2} = 4

= 4 u^{4}

3 u = 4 u^{4}

3 u = 4 u^{4}

3 u = 4 u^{4}

解

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

3 u = 4 u^{4}

将

4 u^{4}

para o lado esquerdo

3 u = 4 u^{4}

两边减去

4 u^{4}

3 u - 4 u^{4} = 4 u^{4} - 4 u^{4}

化简

3 u - 4 u^{4} = 0

3 u - 4 u^{4} = 0

因式分解

3 u - 4 u^{4} : - u (34 u - 33) (4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}})

3 u - 4 u^{4}

因式分解出通项

- u : - u (4 u^{3} - 3)

- 4 u^{4} + 3 u

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{4} = u^{3} u

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

因式分解出通项

- u

= - u (4 u^{3} - 3)

= - u (4 u^{3} - 3)

分解

4 u^{3} - 3 : (34 u - 33) ((34)^{2} u^{2} + 3334 u + (33)^{2})

4 u^{3} - 3

将

4 u^{3} - 3

改写为

(34 u)^{3} - (33)^{3}

4 u^{3} - 3

使用根式运算法则:

a = (a)^{2}

4 = (34)^{3}

= (34)^{3} u^{3} - 3

使用根式运算法则:

a = (a)^{2}

3 = (33)^{3}

= (34)^{3} u^{3} - (33)^{3}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(34)^{3} u^{3} = (34 u)^{3}

= (34 u)^{3} - (33)^{3}

= (34 u)^{3} - (33)^{3}

使用立方差公式:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(34 u)^{3} - (33)^{3} = (34 u - 33) ((34)^{2} u^{2} + 3334 u + (33)^{2})

= (34 u - 33) ((34)^{2} u^{2} + 3334 u + (33)^{2})

= - u (34 u - 33) ((34)^{2} u^{2} + 3334 u + (33)^{2})

整理后得

= - u (34 u - 33) (4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}})

- u (34 u - 33) (4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}}) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 34 u - 33 = 0 or 4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}} = 0

解

34 u - 33 = 0 : u = 3 \frac{3}{4}

34 u - 33 = 0

将

33

到右边

34 u - 33 = 0

两边加上

33

34 u - 33 + 33 = 0 + 33

化简

34 u = 33

34 u = 33

两边除以

34

34 u = 33

两边除以

34

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

化简

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

化简

\frac{3 4 u}{3 4} : u

\frac{3 4 u}{3 4}

约分:

34

= u

化简

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

\frac{3 3}{3 4}

合并相同指数项 :

\frac{n x}{n y} = n \frac{x}{y}

= 3 \frac{3}{4}

u = 3 \frac{3}{4}

u = 3 \frac{3}{4}

u = 3 \frac{3}{4}

解

4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}} = 0 : u \in R

无解

4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}} = 0

判别式

4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}} = 0 : - 3 \cdot 1 2^{\frac{2}{3}}

4^{\frac{2}{3}} u^{2} + 312 u + 3^{\frac{2}{3}} = 0

对于

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

形式的二次方程，根判别式为

b^{2} - 4 a c

若

a = 4^{\frac{2}{3}}, b = 312, c = 3^{\frac{2}{3}} : (312)^{2} - 4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

(312)^{2} - 4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

展开

(312)^{2} - 4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} : - 3 \cdot 1 2^{\frac{2}{3}}

(312)^{2} - 4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

(312)^{2} = 1 2^{\frac{2}{3}}

(312)^{2}

使用根式运算法则:

n a = a^{\frac{1}{n}}

= (1 2^{\frac{1}{3}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 1 2^{\frac{1}{3} \cdot 2}

\frac{1}{3} \cdot 2 = \frac{2}{3}

\frac{1}{3} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{3}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{3}

= 1 2^{\frac{2}{3}}

4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = 4 \cdot 1 2^{\frac{2}{3}}

4 \cdot 4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

4^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = (4 \cdot 3)^{\frac{2}{3}}

= 4 (4 \cdot 3)^{\frac{2}{3}}

数字相乘:

4 \cdot 3 = 12

= 4 \cdot 1 2^{\frac{2}{3}}

= 1 2^{\frac{2}{3}} - 4 \cdot 1 2^{\frac{2}{3}}

同类项相加:

1 2^{\frac{2}{3}} - 4 \cdot 1 2^{\frac{2}{3}} = - 3 \cdot 1 2^{\frac{2}{3}}

= - 3 \cdot 1 2^{\frac{2}{3}}

- 3 \cdot 1 2^{\frac{2}{3}}

判别式在

u

内不能为负

\in R

解是

u \in R 无解

解为

u = 0, u = 3 \frac{3}{4}

u = 0, u = 3 \frac{3}{4}

验证解:

u = 0

真

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

真

将它们代入

2 u^{2} - 3 u = 0

检验解是否符合

去除与方程不符的解。

代入

u = 0 :

真

2 \cdot 0^{2} - 3 \cdot 0 = 0

2 \cdot 0^{2} - 3 \cdot 0 = 0

2 \cdot 0^{2} - 3 \cdot 0

使用法则

0^{a} = 0

0^{2} = 0

= 2 \cdot 0 - 3 \cdot 0

2 \cdot 0 = 0

2 \cdot 0

使用法则

0 \cdot a = 0

= 0

3 \cdot 0 = 0

3 \cdot 0

使用法则

0 \cdot a = 0

= 0

使用法则

0 = 0

= 0

= 0 - 0

数字相减:

0 - 0 = 0

= 0

0 = 0

真

代入

u = 3 \frac{3}{4} :

真

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

2 (3 \frac{3}{4})^{2} - 3 3 \frac{3}{4} = 2 (\frac{3}{4})^{\frac{2}{3}} - 3 6 \frac{3}{4}

2 (3 \frac{3}{4})^{2} - 3 3 \frac{3}{4}

2 (3 \frac{3}{4})^{2} = 2 (\frac{3}{4})^{\frac{2}{3}}

2 (3 \frac{3}{4})^{2}

(3 \frac{3}{4})^{2} = (\frac{3}{4})^{\frac{2}{3}}

(3 \frac{3}{4})^{2}

使用根式运算法则:

n a = a^{\frac{1}{n}}

= ((\frac{3}{4})^{\frac{1}{3}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (\frac{3}{4})^{\frac{1}{3} \cdot 2}

\frac{1}{3} \cdot 2 = \frac{2}{3}

\frac{1}{3} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{3}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{3}

= (\frac{3}{4})^{\frac{2}{3}}

= 2 (\frac{3}{4})^{\frac{2}{3}}

3 3 \frac{3}{4} = 3 6 \frac{3}{4}

3 3 \frac{3}{4}

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 3 3 \frac{3}{4}

3 \frac{3}{4} : 6 \frac{3}{4}

使用根式运算法则:

n a = a^{\frac{1}{n}}

= ((\frac{3}{4})^{\frac{1}{3}})^{\frac{1}{2}}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (\frac{3}{4})^{\frac{1}{3} \cdot \frac{1}{2}}

\frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}

\frac{1}{3} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 1}{3 \cdot 2}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{1}{6}

= (\frac{3}{4})^{\frac{1}{6}}

a^{\frac{1}{n}} = n a

= 6 \frac{3}{4}

= 3 6 \frac{3}{4}

= 2 (\frac{3}{4})^{\frac{2}{3}} - 3 6 \frac{3}{4}

2 (\frac{3}{4})^{\frac{2}{3}} - 3 6 \frac{3}{4} = 0

真

解为

u = 0, u = 3 \frac{3}{4}

u = cos (x)

代回

cos (x) = 0, cos (x) = 3 \frac{3}{4}

cos (x) = 0, cos (x) = 3 \frac{3}{4}

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

的通解

cos (x)

周期表（周期为

2 πn

）：

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}

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

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

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

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

使用反三角函数性质

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

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

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (3 \frac{3}{4}) + 2 πn, x = 2 π - arccos (3 \frac{3}{4}) + 2 πn

x = arccos (3 \frac{3}{4}) + 2 πn, x = 2 π - arccos (3 \frac{3}{4}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arccos (3 \frac{3}{4}) + 2 πn, x = 2 π - arccos (3 \frac{3}{4}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.43097 \dots + 2 πn, x = 2 π - 0.43097 \dots + 2 πn

作图

流行的例子

sin(3x)=sin(5x)

sin (3 x) = sin (5 x)

2sin^2(3θ)+sin(3θ)-1=0

2 sin^{2} (3 θ) + sin (3 θ) - 1 = 0

4cos(θ)+sqrt(3)=2cos(θ)

4 cos (θ) + 3 = 2 cos (θ)

6cos^2(θ)-cos(θ)-1=0

6 cos^{2} (θ) - cos (θ) - 1 = 0

∣ cos (3 x) ∣ = \frac{1}{2}