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受欢迎的三角函数 >

4sin(x)=csc(x)

解答

4 sin (x) = csc (x)

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin (x) = csc (x)

两边减去

csc (x)

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

使用三角恒等式改写

- csc (x) + 4 sin (x)

使用基本三角恒等式:

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

= - csc (x) + 4 \cdot \frac{1}{csc ( x )}

4 \cdot \frac{1}{csc ( x )} = \frac{4}{csc ( x )}

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

分式相乘:

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

= \frac{1 \cdot 4}{csc ( x )}

数字相乘:

1 \cdot 4 = 4

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

= - csc (x) + \frac{4}{csc ( x )}

- csc (x) + \frac{4}{csc ( x )} = 0

用替代法求解

- csc (x) + \frac{4}{csc ( x )} = 0

令

: csc (x) = u

- u + \frac{4}{u} = 0

- u + \frac{4}{u} = 0 : u = 2, u = - 2

- u + \frac{4}{u} = 0

在两边乘以

u

- u + \frac{4}{u} = 0

在两边乘以

u

- uu + \frac{4}{u} u = 0 \cdot u

化简

- uu + \frac{4}{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{4}{u} u : 4

\frac{4}{u} u

分式相乘:

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

= \frac{4 u}{u}

约分:

u

= 4

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- u^{2} + 4 = 0

- u^{2} + 4 = 0

- u^{2} + 4 = 0

解

- u^{2} + 4 = 0 : u = 2, u = - 2

- u^{2} + 4 = 0

将

4

到右边

- u^{2} + 4 = 0

两边减去

4

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

化简

- u^{2} = - 4

- u^{2} = - 4

两边除以

- 1

- u^{2} = - 4

两边除以

- 1

\frac{- u ^{2}}{- 1} = \frac{- 4}{- 1}

化简

u^{2} = 4

u^{2} = 4

对于

x^{2} = f (a)

解为

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

u = 4, u = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= - 2

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{4}{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}{6} + 2 πn, x = \frac{5 π}{6} + 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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

流行的例子

cot (θ) = - \frac{3}{2}

7sin(x)+sqrt(23)=0

7 sin (x) + 23 = 0

4cos^2(x)+cos(x)-2=0

4 cos^{2} (x) + cos (x) - 2 = 0

tan(x)=-1+sqrt(2)

tan (x) = - 1 + 2

cot (θ) = - \frac{3}{4}