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受欢迎的三角函数 >

3cot^2(x)=7csc(x)-7

解答

3 cot^{2} (x) = 7 csc (x) - 7

解答

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

+1

度数

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

3 cot^{2} (x) = 7 csc (x) - 7

两边减去

7 csc (x) - 7

3 cot^{2} (x) - 7 csc (x) + 7 = 0

使用三角恒等式改写

7 + 3 cot^{2} (x) - 7 csc (x)

使用毕达哥拉斯恒等式:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= 7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

化简

7 + 3 (csc^{2} (x) - 1) - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

乘开

3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 3

3 (csc^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

= 3 csc^{2} (x) - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 3 csc^{2} (x) - 3

= 7 + 3 csc^{2} (x) - 3 - 7 csc (x)

化简

7 + 3 csc^{2} (x) - 3 - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 csc^{2} (x) - 3 - 7 csc (x)

对同类项分组

= 3 csc^{2} (x) - 7 csc (x) + 7 - 3

数字相加/相减:

7 - 3 = 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

4 + 3 csc^{2} (x) - 7 csc (x) = 0

用替代法求解

4 + 3 csc^{2} (x) - 7 csc (x) = 0

令

: csc (x) = u

4 + 3 u^{2} - 7 u = 0

4 + 3 u^{2} - 7 u = 0 : u = \frac{4}{3}, u = 1

4 + 3 u^{2} - 7 u = 0

改写成标准形式

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

3 u^{2} - 7 u + 4 = 0

使用求根公式求解

3 u^{2} - 7 u + 4 = 0

二次方程求根公式:

若

a = 3, b = - 7, c = 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

u = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

数字相加:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{8}{6}

约分:

2

= \frac{4}{3}

u = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

数字相减:

7 - 1 = 6

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

数字相乘:

2 \cdot 3 = 6

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{4}{3}, u = 1

u = csc (x)

代回

csc (x) = \frac{4}{3}, csc (x) = 1

csc (x) = \frac{4}{3}, csc (x) = 1

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

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

使用反三角函数性质

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

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

的通解

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

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

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

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

csc (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

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

以小数形式表示解

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

作图

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