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受欢迎的三角函数 >

2sin^2(x)=(csc^2(x))/2

解答

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

解答

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

度数

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

求解步骤

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

两边减去

\frac{csc ^{2} ( x )}{2}

2 sin^{2} (x) - \frac{csc ^{2} ( x )}{2} = 0

化简

2 sin^{2} (x) - \frac{csc ^{2} ( x )}{2} : \frac{4 sin ^{2} ( x ) - csc ^{2} ( x )}{2}

2 sin^{2} (x) - \frac{csc ^{2} ( x )}{2}

将项转换为分式:

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

= \frac{2 sin ^{2} ( x ) \cdot 2}{2} - \frac{csc ^{2} ( x )}{2}

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 sin ^{2} ( x ) \cdot 2 - csc ^{2} ( x )}{2}

数字相乘:

2 \cdot 2 = 4

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

\frac{4 sin ^{2} ( x ) - csc ^{2} ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

分解

4 sin^{2} (x) - csc^{2} (x) : (2 sin (x) + csc (x)) (2 sin (x) - csc (x))

4 sin^{2} (x) - csc^{2} (x)

将

4 sin^{2} (x) - csc^{2} (x)

改写为

(2 sin (x))^{2} - csc^{2} (x)

4 sin^{2} (x) - csc^{2} (x)

将

4

改写为

2^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

(2 sin (x))^{2} - csc^{2} (x) = (2 sin (x) + csc (x)) (2 sin (x) - csc (x))

= (2 sin (x) + csc (x)) (2 sin (x) - csc (x))

(2 sin (x) + csc (x)) (2 sin (x) - csc (x)) = 0

分别求解每个部分

2 sin (x) + csc (x) = 0 or 2 sin (x) - csc (x) = 0

2 sin (x) + csc (x) = 0 :

无解

2 sin (x) + csc (x) = 0

使用三角恒等式改写

csc (x) + 2 sin (x)

使用基本三角恒等式:

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

= csc (x) + 2 \cdot \frac{1}{csc ( x )}

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

2 \cdot \frac{1}{csc ( x )}

分式相乘:

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

= \frac{1 \cdot 2}{csc ( x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{csc ( x )}

= csc (x) + \frac{2}{csc ( x )}

csc (x) + \frac{2}{csc ( x )} = 0

用替代法求解

csc (x) + \frac{2}{csc ( x )} = 0

令

: csc (x) = u

u + \frac{2}{u} = 0

u + \frac{2}{u} = 0 : u = 2 i, u = - 2 i

u + \frac{2}{u} = 0

在两边乘以

u

u + \frac{2}{u} = 0

在两边乘以

u

uu + \frac{2}{u} u = 0 \cdot u

化简

uu + \frac{2}{u} u = 0 \cdot u

化简

uu : u^{2}

uu

使用指数法则:

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

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} + 2 = 0

u^{2} + 2 = 0

u^{2} + 2 = 0

解

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

将

2

到右边

u^{2} + 2 = 0

两边减去

2

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

化简

u^{2} = - 2

u^{2} = - 2

对于

x^{2} = f (a)

解为

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

u = - 2, u = - - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

化简

- - 2 : - 2 i

- - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

u = 2 i, u = - 2 i

u = csc (x)

代回

csc (x) = 2 i, csc (x) = - 2 i

csc (x) = 2 i, csc (x) = - 2 i

csc (x) = 2 i :

无解

csc (x) = 2 i

无解

csc (x) = - 2 i :

无解

csc (x) = - 2 i

无解

合并所有解

无解

2 sin (x) - csc (x) = 0 : 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

2 sin (x) - csc (x) = 0

使用三角恒等式改写

- csc (x) + 2 sin (x)

使用基本三角恒等式:

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

= - csc (x) + 2 \cdot \frac{1}{csc ( x )}

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

2 \cdot \frac{1}{csc ( x )}

分式相乘:

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

= \frac{1 \cdot 2}{csc ( x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{csc ( x )}

= - csc (x) + \frac{2}{csc ( x )}

- csc (x) + \frac{2}{csc ( x )} = 0

用替代法求解

- csc (x) + \frac{2}{csc ( x )} = 0

令

: csc (x) = u

- u + \frac{2}{u} = 0

- u + \frac{2}{u} = 0 : u = 2, u = - 2

- u + \frac{2}{u} = 0

在两边乘以

u

- u + \frac{2}{u} = 0

在两边乘以

u

- uu + \frac{2}{u} u = 0 \cdot u

化简

- uu + \frac{2}{u} u = 0 \cdot u

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- u^{2} + 2 = 0

- u^{2} + 2 = 0

- u^{2} + 2 = 0

解

- u^{2} + 2 = 0 : u = 2, u = - 2

- u^{2} + 2 = 0

将

2

到右边

- u^{2} + 2 = 0

两边减去

2

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

化简

- u^{2} = - 2

- u^{2} = - 2

两边除以

- 1

- u^{2} = - 2

两边除以

- 1

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

化简

u^{2} = 2

u^{2} = 2

对于

x^{2} = f (a)

解为

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

u = 2, u = - 2

u = 2, u = - 2

验证解

找到无定义的点（奇点）:

u = 0

取

- u + \frac{2}{u}

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 2, u = - 2

u = csc (x)

代回

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

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

csc (x) = 2

csc (x) = 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 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

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

csc (x) = - 2

csc (x) = - 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 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

合并所有解

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

合并所有解

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

作图

流行的例子

5 cos (t) = 0

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

cos (x) = \frac{6}{10}

sec^2(x)-2tan^2(x)= 2/3

sec^{2} (x) - 2 tan^{2} (x) = \frac{2}{3}

cos^2(x)-3cos(x)-1=0

cos^{2} (x) - 3 cos (x) - 1 = 0