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受欢迎的三角函数 >

3tan(x)+cot(x)-5csc(x)=0

解答

3 tan (x) + cot (x) - 5 csc (x) = 0

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

3 tan (x) + cot (x) - 5 csc (x) = 0

用 sin, cos 表示

cot (x) + 3 tan (x) - 5 csc (x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + 3 tan (x) - 5 csc (x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 csc (x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )}

化简

\frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )} : \frac{cos ( x ) ( cos ( x ) - 5 ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )}

3 \cdot \frac{sin ( x )}{cos ( x )} = \frac{3 sin ( x )}{cos ( x )}

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

分式相乘:

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

= \frac{sin ( x ) \cdot 3}{cos ( x )}

5 \cdot \frac{1}{sin ( x )} = \frac{5}{sin ( x )}

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

分式相乘:

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

= \frac{1 \cdot 5}{sin ( x )}

数字相乘:

1 \cdot 5 = 5

= \frac{5}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} + \frac{3 sin ( x )}{cos ( x )} - \frac{5}{sin ( x )}

合并分式

\frac{cos ( x )}{sin ( x )} - \frac{5}{sin ( x )} : \frac{cos ( x ) - 5}{sin ( x )}

使用法则

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

= \frac{cos ( x ) - 5}{sin ( x )}

= \frac{cos ( x ) - 5}{sin ( x )} + \frac{3 sin ( x )}{cos ( x )}

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

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

对于

\frac{cos ( x ) - 5}{sin ( x )} :

将分母和分子乘以

cos (x)

\frac{cos ( x ) - 5}{sin ( x )} = \frac{( cos ( x ) - 5 ) cos ( x )}{sin ( x ) cos ( x )}

对于

\frac{sin ( x ) \cdot 3}{cos ( x )} :

将分母和分子乘以

sin (x)

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

= \frac{( cos ( x ) - 5 ) cos ( x )}{sin ( x ) cos ( x )} + \frac{3 sin ^{2} ( x )}{sin ( x ) cos ( 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 ) - 5 ) cos ( x ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{cos ( x ) ( cos ( x ) - 5 ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{( - 5 + cos ( x ) ) cos ( x ) + 3 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

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

使用三角恒等式改写

(- 5 + cos (x)) cos (x) + 3 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)

= (- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x))

化简

(- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x)) : - 5 cos (x) - 2 cos^{2} (x) + 3

(- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x))

= cos (x) (- 5 + cos (x)) + 3 (1 - cos^{2} (x))

乘开

cos (x) (- 5 + cos (x)) : - 5 cos (x) + cos^{2} (x)

cos (x) (- 5 + cos (x))

使用分配律:

a (b + c) = ab + a c

a = cos (x), b = - 5, c = cos (x)

= cos (x) (- 5) + cos (x) cos (x)

使用加减运算法则

+ (- a) = - a

= - 5 cos (x) + cos (x) cos (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)

= - 5 cos (x) + cos^{2} (x)

= - 5 cos (x) + cos^{2} (x) + 3 (1 - cos^{2} (x))

乘开

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

数字相乘:

3 \cdot 1 = 3

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

= - 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x)

化简

- 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x) : - 5 cos (x) - 2 cos^{2} (x) + 3

- 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x)

对同类项分组

= - 5 cos (x) + cos^{2} (x) - 3 cos^{2} (x) + 3

同类项相加:

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

= - 5 cos (x) - 2 cos^{2} (x) + 3

= - 5 cos (x) - 2 cos^{2} (x) + 3

= - 5 cos (x) - 2 cos^{2} (x) + 3

3 - 2 cos^{2} (x) - 5 cos (x) = 0

用替代法求解

3 - 2 cos^{2} (x) - 5 cos (x) = 0

令

: cos (x) = u

3 - 2 u^{2} - 5 u = 0

3 - 2 u^{2} - 5 u = 0 : u = - 3, u = \frac{1}{2}

3 - 2 u^{2} - 5 u = 0

改写成标准形式

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

- 2 u^{2} - 5 u + 3 = 0

使用求根公式求解

- 2 u^{2} - 5 u + 3 = 0

二次方程求根公式:

若

a = - 2, b = - 5, c = 3

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

(- 5)^{2} - 4 (- 2) \cdot 3 = 7

(- 5)^{2} - 4 (- 2) \cdot 3

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 2 \cdot 3

使用指数法则:

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

若

n

是偶数

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

= 5^{2} + 4 \cdot 2 \cdot 3

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 7}{2 ( - 2 )}, u_{2} = \frac{- ( - 5 ) - 7}{2 ( - 2 )}

u = \frac{- ( - 5 ) + 7}{2 ( - 2 )} : - 3

\frac{- ( - 5 ) + 7}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{5 + 7}{- 2 \cdot 2}

数字相加:

5 + 7 = 12

= \frac{12}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{12}{- 4}

使用分式法则:

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

= - \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= - 3

u = \frac{- ( - 5 ) - 7}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 5 ) - 7}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{5 - 7}{- 2 \cdot 2}

数字相减:

5 - 7 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

使用分式法则:

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

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

u = - 3, u = \frac{1}{2}

u = cos (x)

代回

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

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

cos (x) = - 3 :

无解

cos (x) = - 3

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

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}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

合并所有解

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

3cos^2(x)=1-5sin(x)

3 cos^{2} (x) = 1 - 5 sin (x)

solvefor x,2sin(3x)-1=0

so l v e f or x, 2 sin (3 x) - 1 = 0

2(cos(t))2-cos(t)-1=0

2 (cos (t)) 2 - cos (t) - 1 = 0

cos(c)=cos(7)cos(60)

cos (c) = cos (7) cos (6 0^{\circ})

cos(xpi)=(-1)^n

cos (x π) = (- 1)^{n}