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受欢迎的三角函数 >

csc(x)-sin(x)=cot(x)*csc(x)

解答

csc (x) - sin (x) = cot (x) \cdot csc (x)

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

csc (x) - sin (x) = cot (x) csc (x)

两边减去

cot (x) csc (x)

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

用 sin, cos 表示

csc (x) - sin (x) - cot (x) csc (x)

使用基本三角恒等式:

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

= \frac{1}{sin ( x )} - sin (x) - cot (x) \frac{1}{sin ( x )}

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1}{sin ( x )} - sin (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

化简

\frac{1}{sin ( x )} - sin (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : \frac{sin ( x ) - sin ^{3} ( x ) - cos ( x )}{sin ^{2} ( x )}

\frac{1}{sin ( x )} - sin (x) - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} = \frac{cos ( x )}{sin ^{2} ( x )}

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

分式相乘:

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

= \frac{cos ( x ) \cdot 1}{sin ( x ) sin ( x )}

乘以:

cos (x) \cdot 1 = cos (x)

= \frac{cos ( x )}{sin ( x ) sin ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

= \frac{cos ( x )}{sin ^{2} ( x )}

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

将项转换为分式:

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

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

sin (x), 1, sin^{2} (x)

的最小公倍数:

sin^{2} (x)

sin (x), 1, sin^{2} (x)

最小公倍数 (LCM)

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= sin^{2} (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin^{2} (x)

对于

\frac{1}{sin ( x )} :

将分母和分子乘以

sin (x)

\frac{1}{sin ( x )} = \frac{1 \cdot sin ( x )}{sin ( x ) sin ( x )} = \frac{sin ( x )}{sin ^{2} ( x )}

对于

\frac{sin ( x )}{1} :

将分母和分子乘以

sin^{2} (x)

\frac{sin ( x )}{1} = \frac{sin ( x ) sin ^{2} ( x )}{1 \cdot sin ^{2} ( x )} = \frac{sin ^{3} ( x )}{sin ^{2} ( x )}

= \frac{sin ( x )}{sin ^{2} ( x )} - \frac{sin ^{3} ( x )}{sin ^{2} ( x )} - \frac{cos ( x )}{sin ^{2} ( x )}

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{sin ( x ) - sin ^{3} ( x ) - cos ( x )}{sin ^{2} ( x )}

= \frac{sin ( x ) - sin ^{3} ( x ) - cos ( x )}{sin ^{2} ( x )}

\frac{- cos ( x ) + sin ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )} = 0

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

- cos (x) + sin (x) - sin^{3} (x) = 0

使用指数法则:

a^{b} = a^{2} a^{b - 2}

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

使用三角恒等式改写

- cos (x) + sin (x) - sin (x) sin^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - cos (x) + sin (x) - sin (x) (1 - cos^{2} (x))

化简

- cos (x) + sin (x) - sin (x) (1 - cos^{2} (x)) : - cos (x) + cos^{2} (x) sin (x)

- cos (x) + sin (x) - sin (x) (1 - cos^{2} (x))

乘开

- sin (x) (1 - cos^{2} (x)) : - sin (x) + cos^{2} (x) sin (x)

- sin (x) (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - sin (x), b = 1, c = cos^{2} (x)

= - sin (x) \cdot 1 - (- sin (x)) cos^{2} (x)

使用加减运算法则

- (- a) = a

= - 1 \cdot sin (x) + cos^{2} (x) sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= - sin (x) + cos^{2} (x) sin (x)

= - cos (x) + sin (x) - sin (x) + cos^{2} (x) sin (x)

同类项相加:

sin (x) - sin (x) = 0

= - cos (x) + cos^{2} (x) sin (x)

= - cos (x) + cos^{2} (x) sin (x)

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

分解

- cos (x) + cos^{2} (x) sin (x) : cos (x) (- 1 + sin (x) cos (x))

- cos (x) + cos^{2} (x) sin (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

sin (x) cos^{2} (x) = cos (x) cos (x)

= - cos (x) + cos (x) cos (x)

因式分解出通项

cos (x)

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

cos (x) (- 1 + sin (x) cos (x)) = 0

分别求解每个部分

cos (x) = 0 or - 1 + sin (x) cos (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

- 1 + sin (x) cos (x) = 0 :

无解

- 1 + sin (x) cos (x) = 0

使用三角恒等式改写

- 1 + sin (x) cos (x)

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

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

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

将

1

到右边

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

两边加上

1

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

化简

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

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

在两边乘以

2

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

在两边乘以

2

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

流行的例子

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

1=sin(t)+sqrt(3)cos(t)

1 = sin (t) + 3 cos (t)

cosh (x) = \frac{5}{4}

sec(θ)-sqrt(2)tan(θ)=0

sec (θ) - 2 tan (θ) = 0

2sin(x-pi/3)=-sqrt(2)

2 sin (x - \frac{π}{3}) = - 2