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受欢迎的三角函数 >

csc^2(x)=5csc(x)

解答

csc^{2} (x) = 5 csc (x)

解答

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

+1

度数

x = 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 168.46304 \dots^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (x) = 5 csc (x)

用替代法求解

csc^{2} (x) = 5 csc (x)

令

: csc (x) = u

u^{2} = 5 u

u^{2} = 5 u : u = 5, u = 0

u^{2} = 5 u

将

5 u

para o lado esquerdo

u^{2} = 5 u

两边减去

5 u

u^{2} - 5 u = 5 u - 5 u

化简

u^{2} - 5 u = 0

u^{2} - 5 u = 0

使用求根公式求解

u^{2} - 5 u = 0

二次方程求根公式:

若

a = 1, b = - 5, c = 0

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot 0 = 5

(- 5)^{2} - 4 \cdot 1 \cdot 0

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 5

u_{1, 2} = \frac{- ( - 5 ) \pm 5}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 5}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 5}{2 \cdot 1}

u = \frac{- ( - 5 ) + 5}{2 \cdot 1} : 5

\frac{- ( - 5 ) + 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{5 + 5}{2 \cdot 1}

数字相加:

5 + 5 = 10

= \frac{10}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{10}{2}

数字相除:

\frac{10}{2} = 5

= 5

u = \frac{- ( - 5 ) - 5}{2 \cdot 1} : 0

\frac{- ( - 5 ) - 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{5 - 5}{2 \cdot 1}

数字相减:

5 - 5 = 0

= \frac{0}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{0}{2}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

二次方程组的解是:

u = 5, u = 0

u = csc (x)

代回

csc (x) = 5, csc (x) = 0

csc (x) = 5, csc (x) = 0

csc (x) = 5 : x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 5

使用反三角函数性质

csc (x) = 5

csc (x) = 5

的通解

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

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

csc (x) = 0 :

无解

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

无解

合并所有解

x = arccsc (5) + 2 πn, x = π - arccsc (5) + 2 πn

以小数形式表示解

x = 0.20135 \dots + 2 πn, x = π - 0.20135 \dots + 2 πn

作图

流行的例子

cos (4 θ) = 0

solvefor t,L{sin(2t)}= 2/(s^2+4)

so l v e f or t, L {sin (2 t)} = \frac{2}{s ^{2} + 4}

2sec^2(x)+tan(x)=5

2 sec^{2} (x) + tan (x) = 5

1-sin(x)=cos(2x)

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sin (x) - \frac{1}{2} = 0