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受欢迎的三角函数 >

3sin(x)+csc(x)=0

解答

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

解答

x \in R 无解

求解步骤

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

使用三角恒等式改写

csc (x) + 3 sin (x)

使用基本三角恒等式:

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

= csc (x) + 3 \cdot \frac{1}{csc ( x )}

3 \cdot \frac{1}{csc ( x )} = \frac{3}{csc ( x )}

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

分式相乘:

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

= \frac{1 \cdot 3}{csc ( x )}

数字相乘:

1 \cdot 3 = 3

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

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

csc (x) + \frac{3}{csc ( x )} = 0

用替代法求解

csc (x) + \frac{3}{csc ( x )} = 0

令

: csc (x) = u

u + \frac{3}{u} = 0

u + \frac{3}{u} = 0 : u = 3 i, u = - 3 i

u + \frac{3}{u} = 0

在两边乘以

u

u + \frac{3}{u} = 0

在两边乘以

u

uu + \frac{3}{u} u = 0 \cdot u

化简

uu + \frac{3}{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{3}{u} u : 3

\frac{3}{u} u

分式相乘:

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

= \frac{3 u}{u}

约分:

u

= 3

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} + 3 = 0

u^{2} + 3 = 0

u^{2} + 3 = 0

解

u^{2} + 3 = 0 : u = 3 i, u = - 3 i

u^{2} + 3 = 0

将

3

到右边

u^{2} + 3 = 0

两边减去

3

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

化简

u^{2} = - 3

u^{2} = - 3

对于

x^{2} = f (a)

解为

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

u = - 3, u = - - 3

化简

- 3 : 3 i

- 3

使用根式运算法则:

- a = - 1 a

- 3 = - 1 3

= - 1 3

使用虚数运算法则:

- 1 = i

= 3 i

化简

- - 3 : - 3 i

- - 3

化简

- 3 : 3 i

- 3

使用根式运算法则:

- a = - 1 a

- 3 = - 1 3

= - 1 3

使用虚数运算法则:

- 1 = i

= 3 i

= - 3 i

u = 3 i, u = - 3 i

u = 3 i, u = - 3 i

u = csc (x)

代回

csc (x) = 3 i, csc (x) = - 3 i

csc (x) = 3 i, csc (x) = - 3 i

csc (x) = 3 i :

无解

csc (x) = 3 i

无解

csc (x) = - 3 i :

无解

csc (x) = - 3 i

无解

合并所有解

x \in R 无解

作图

流行的例子

cos (x) = \frac{4}{7}

csc^2(x)-2cot(x)=0

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

tan (θ) = - \frac{4}{3}

5 csc (θ) - 3 = 2

3cos(θ-55)=-1.56

3 cos (θ - 5 5^{\circ}) = - 1.56