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受欢迎的三角函数 >

1/(csc^2(a))=1-sin^2(a)

解答

\frac{1}{csc ^{2} ( a )} = 1 - sin^{2} (a)

解答

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

+1

度数

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

求解步骤

\frac{1}{csc ^{2} ( a )} = 1 - sin^{2} (a)

两边减去

1 - sin^{2} (a)

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a) = 0

化简

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a) : \frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a)

将项转换为分式:

1 = \frac{1 csc ^{2} ( a )}{csc ^{2} ( a )}, sin^{2} (a) = \frac{sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

= \frac{1}{csc ^{2} ( a )} - \frac{1 \cdot csc ^{2} ( a )}{csc ^{2} ( a )} + \frac{sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

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{1 - 1 \cdot csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

乘以:

1 \cdot csc^{2} (a) = csc^{2} (a)

= \frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

\frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )} = 0

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

1 - csc^{2} (a) + sin^{2} (a) csc^{2} (a) = 0

使用三角恒等式改写

1 - csc^{2} (a) + csc^{2} (a) sin^{2} (a)

使用基本三角恒等式:

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

= 1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2}

化简

1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2} : - csc^{2} (a) + 2

1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2}

csc^{2} (a) (\frac{1}{csc ( a )})^{2} = 1

csc^{2} (a) (\frac{1}{csc ( a )})^{2}

(\frac{1}{csc ( a )})^{2} = \frac{1}{csc ^{2} ( a )}

(\frac{1}{csc ( a )})^{2}

使用指数法则:

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

= \frac{1 ^{2}}{csc ^{2} ( a )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( a )}

= \frac{1}{csc ^{2} ( a )} csc^{2} (a)

分式相乘:

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

= \frac{1 \cdot csc ^{2} ( a )}{csc ^{2} ( a )}

约分:

csc^{2} (a)

= 1

= 1 - csc^{2} (a) + 1

对同类项分组

= - csc^{2} (a) + 1 + 1

数字相加:

1 + 1 = 2

= - csc^{2} (a) + 2

= - csc^{2} (a) + 2

2 - csc^{2} (a) = 0

用替代法求解

2 - csc^{2} (a) = 0

令

: csc (a) = u

2 - u^{2} = 0

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

2 - u^{2} = 0

将

2

到右边

2 - u^{2} = 0

两边减去

2

2 - u^{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 = csc (a)

代回

csc (a) = 2, csc (a) = - 2

csc (a) = 2, csc (a) = - 2

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

csc (a) = 2

csc (a) = 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

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

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

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

csc (a) = - 2

csc (a) = - 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

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

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

合并所有解

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

作图

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