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受欢迎的三角函数 >

2sin^2(x)-sqrt(2sin(x))=0

解答

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

解答

x = 2 πn, x = π + 2 πn, x = 0.91686 \dots + 2 πn, x = π - 0.91686 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 52.53268 \dots^{\circ} + 36 0^{\circ} n, x = 127.46731 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

2 u^{2} - 2 u = 0

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

2 u^{2} - 2 u = 0

去除平方根

2 u^{2} - 2 u = 0

两边减去

2 u^{2}

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

化简

- 2 u = - 2 u^{2}

两边进行平方:

2 u = 4 u^{4}

2 u^{2} - 2 u = 0

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

展开

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

(- 2 u)^{2}

使用指数法则:

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

若

n

是偶数

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

= (2 u)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

= (2 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

= 2 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}

2 u = 4 u^{4}

2 u = 4 u^{4}

2 u = 4 u^{4}

解

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

2 u = 4 u^{4}

将

4 u^{4}

para o lado esquerdo

2 u = 4 u^{4}

两边减去

4 u^{4}

2 u - 4 u^{4} = 4 u^{4} - 4 u^{4}

化简

2 u - 4 u^{4} = 0

2 u - 4 u^{4} = 0

因式分解

2 u - 4 u^{4} : - 2 u (32 u - 1) (2^{\frac{2}{3}} u^{2} + 32 u + 1)

2 u - 4 u^{4}

因式分解出通项

- 2 u : - 2 u (2 u^{3} - 1)

- 4 u^{4} + 2 u

使用指数法则:

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

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

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

将

4

改写为

2 \cdot 2

= - 2 \cdot 2 u^{3} u + 2 u

因式分解出通项

- 2 u

= - 2 u (2 u^{3} - 1)

= - 2 u (2 u^{3} - 1)

分解

2 u^{3} - 1 : (32 u - 1) ((32)^{2} u^{2} + 32 u + 1)

2 u^{3} - 1

将

2 u^{3} - 1

改写为

(32 u)^{3} - 1^{3}

2 u^{3} - 1

使用根式运算法则:

a = (a)^{2}

2 = (32)^{3}

= (32)^{3} u^{3} - 1

将

1

改写为

1^{3}

= (32)^{3} u^{3} - 1^{3}

使用指数法则:

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

(32)^{3} u^{3} = (32 u)^{3}

= (32 u)^{3} - 1^{3}

= (32 u)^{3} - 1^{3}

使用立方差公式:

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

(32 u)^{3} - 1^{3} = (32 u - 1) ((32)^{2} u^{2} + 32 u + 1)

= (32 u - 1) ((32)^{2} u^{2} + 32 u + 1)

= - 2 u (32 u - 1) ((32)^{2} u^{2} + 32 u + 1)

整理后得

= - 2 u (32 u - 1) (2^{\frac{2}{3}} u^{2} + 32 u + 1)

- 2 u (32 u - 1) (2^{\frac{2}{3}} u^{2} + 32 u + 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 32 u - 1 = 0 or 2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0

解

32 u - 1 = 0 : u = \frac{2 ^{\frac{2}{3}}}{2}

32 u - 1 = 0

将

1

到右边

32 u - 1 = 0

两边加上

1

32 u - 1 + 1 = 0 + 1

化简

32 u = 1

32 u = 1

两边除以

32

32 u = 1

两边除以

32

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

化简

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

化简

\frac{3 2 u}{3 2} : u

\frac{3 2 u}{3 2}

约分:

32

= u

化简

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

\frac{1}{3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

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

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

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

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

使用指数法则:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

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

化简

\frac{2}{3} + \frac{1}{3} : 1

\frac{2}{3} + \frac{1}{3}

因为分母相等，所以合并分式:

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

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

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

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

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

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

解

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0 : u \in R

无解

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0

判别式

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0 : - 3 \cdot 2^{\frac{2}{3}}

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 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 = 2^{\frac{2}{3}}, b = 32, c = 1 : (32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

(32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

展开

(32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1 : - 3 \cdot 2^{\frac{2}{3}}

(32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

(32)^{2} = 2^{\frac{2}{3}}

(32)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

= 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}

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

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

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

数字相乘:

4 \cdot 1 = 4

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

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

同类项相加:

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

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

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

u = 0, u = \frac{2 ^{\frac{2}{3}}}{2}

验证解:

u = 0

真

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

真

将它们代入

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

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

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

使用法则

0^{a} = 0

0^{2} = 0

= 2 \cdot 0 - 2 \cdot 0

2 \cdot 0 = 0

2 \cdot 0

使用法则

0 \cdot a = 0

= 0

2 \cdot 0 = 0

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

真

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

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

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

去除括号:

(a) = a

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

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

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

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

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

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

\frac{2 ^{\frac{2}{3}}}{2}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

数字相减:

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

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

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

使用指数法则:

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

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

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

使用指数法则:

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

= 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}

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

使用指数法则:

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

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

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

数字相减:

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

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

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

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

乘

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

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

分式相乘:

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

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

约分:

2

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

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

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

2^{\frac{1}{3}} - 2^{\frac{2}{3}} = 0

真

解为

u = 0, u = \frac{2 ^{\frac{2}{3}}}{2}

u = sin (x)

代回

sin (x) = 0, sin (x) = \frac{2 ^{\frac{2}{3}}}{2}

sin (x) = 0, sin (x) = \frac{2 ^{\frac{2}{3}}}{2}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (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} 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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = \frac{2 ^{\frac{2}{3}}}{2} : x = arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = π - arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

x = arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = π - arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

x = arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = π - arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = π - arcsin (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 0.91686 \dots + 2 πn, x = π - 0.91686 \dots + 2 πn

作图

流行的例子

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