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受欢迎的三角函数 >

((cos(x)cot(x)))/((1-sin(x))-1)=csc(x)

解答

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

解答

x = - 0.66623 \dots + 2 πn, x = π + 0.66623 \dots + 2 πn

+1

度数

x = - 38.17270 \dots^{\circ} + 36 0^{\circ} n, x = 218.17270 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

csc (x)

- \frac{cos ( x ) cot ( x )}{sin ( x )} - csc (x) = 0

化简

- \frac{cos ( x ) cot ( x )}{sin ( x )} - csc (x) : \frac{- cos ( x ) cot ( x ) - csc ( x ) sin ( x )}{sin ( x )}

- \frac{cos ( x ) cot ( x )}{sin ( x )} - csc (x)

将项转换为分式:

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

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

\frac{- cos ( x ) cot ( x ) - csc ( x ) sin ( x )}{sin ( x )} = 0

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

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

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

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

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

分式相乘:

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

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

使用基本三角恒等式:

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

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

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

分解

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

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

因式分解出通项

csc (x)

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

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

分别求解每个部分

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

csc (x) = 0 :

无解

csc (x) = 0

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

无解

cos^{2} (x) + sin (x) = 0 : x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

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

用替代法求解

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

令

: sin (x) = u

1 + u - u^{2} = 0

1 + u - u^{2} = 0 : u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

1 + u - u^{2} = 0

改写成标准形式

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

- u^{2} + u + 1 = 0

使用求根公式求解

- u^{2} + u + 1 = 0

二次方程求根公式:

若

a = - 1, b = 1, c = 1

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

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

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

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 1

使用法则

- (- a) = a

= 1 + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 1 + 4

数字相加:

1 + 4 = 5

= 5

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

将解分隔开

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

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

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

去除括号:

(- a) = - a

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

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

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

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

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

去除括号:

(- a) = - a

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

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

- 1 - 5 = - (1 + 5)

= \frac{1 + 5}{2}

二次方程组的解是:

u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

u = sin (x)

代回

sin (x) = - \frac{- 1 + 5}{2}, sin (x) = \frac{1 + 5}{2}

sin (x) = - \frac{- 1 + 5}{2}, sin (x) = \frac{1 + 5}{2}

sin (x) = - \frac{- 1 + 5}{2} : x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

sin (x) = - \frac{- 1 + 5}{2}

使用反三角函数性质

sin (x) = - \frac{- 1 + 5}{2}

sin (x) = - \frac{- 1 + 5}{2}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

sin (x) = \frac{1 + 5}{2} :

无解

sin (x) = \frac{1 + 5}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

合并所有解

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

以小数形式表示解

x = - 0.66623 \dots + 2 πn, x = π + 0.66623 \dots + 2 πn

作图

流行的例子

-9sin^2(θ)+cos(θ)+19=-5cos(θ)+9

- 9 sin^{2} (θ) + cos (θ) + 19 = - 5 cos (θ) + 9

4cos(θ)sin(θ)+3cos(θ)=0

4 cos (θ) sin (θ) + 3 cos (θ) = 0

2 + 3 cos (θ) = 0

sin(θ)=-24/25 ,pi<θ<(3pi)/2

sin (θ) = - \frac{24}{25}, π < θ < \frac{3 π}{2}

3 sec^{2} (x) - 6 = 0