zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

6tan(x)-5csc(x)=0

解答

6 tan (x) - 5 csc (x) = 0

解答

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度数

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

6 tan (x) - 5 csc (x) = 0

用 sin, cos 表示

- 5 csc (x) + 6 tan (x)

使用基本三角恒等式:

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

= - 5 \cdot \frac{1}{sin ( x )} + 6 tan (x)

使用基本三角恒等式:

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

= - 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{cos ( x )}

化简

- 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{cos ( x )} : \frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

- 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{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 )}

6 \cdot \frac{sin ( x )}{cos ( x )} = \frac{6 sin ( x )}{cos ( x )}

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

分式相乘:

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

= \frac{sin ( x ) \cdot 6}{cos ( x )}

= - \frac{5}{sin ( x )} + \frac{6 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{5}{sin ( x )} :

将分母和分子乘以

cos (x)

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

对于

\frac{sin ( x ) \cdot 6}{cos ( x )} :

将分母和分子乘以

sin (x)

\frac{sin ( x ) \cdot 6}{cos ( x )} = \frac{sin ( x ) \cdot 6 sin ( x )}{cos ( x ) sin ( x )} = \frac{6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= - \frac{5 cos ( x )}{sin ( x ) cos ( x )} + \frac{6 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{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

- 5 cos (x) + 6 sin^{2} (x) = 0

使用三角恒等式改写

- 5 cos (x) + 6 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) + 6 (1 - cos^{2} (x))

(1 - cos^{2} (x)) \cdot 6 - 5 cos (x) = 0

用替代法求解

(1 - cos^{2} (x)) \cdot 6 - 5 cos (x) = 0

令

: cos (x) = u

(1 - u^{2}) \cdot 6 - 5 u = 0

(1 - u^{2}) \cdot 6 - 5 u = 0 : u = - \frac{3}{2}, u = \frac{2}{3}

(1 - u^{2}) \cdot 6 - 5 u = 0

展开

(1 - u^{2}) \cdot 6 - 5 u : 6 - 6 u^{2} - 5 u

(1 - u^{2}) \cdot 6 - 5 u

= 6 (1 - u^{2}) - 5 u

乘开

6 (1 - u^{2}) : 6 - 6 u^{2}

6 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 6, b = 1, c = u^{2}

= 6 \cdot 1 - 6 u^{2}

数字相乘:

6 \cdot 1 = 6

= 6 - 6 u^{2}

= 6 - 6 u^{2} - 5 u

6 - 6 u^{2} - 5 u = 0

改写成标准形式

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

- 6 u^{2} - 5 u + 6 = 0

使用求根公式求解

- 6 u^{2} - 5 u + 6 = 0

二次方程求根公式:

若

a = - 6, b = - 5, c = 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 6}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 6}{2 ( - 6 )}

(- 5)^{2} - 4 (- 6) \cdot 6 = 13

(- 5)^{2} - 4 (- 6) \cdot 6

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 6 \cdot 6

使用指数法则:

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

若

n

是偶数

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

= 5^{2} + 4 \cdot 6 \cdot 6

数字相乘:

4 \cdot 6 \cdot 6 = 144

= 5^{2} + 144

5^{2} = 25

= 25 + 144

数字相加:

25 + 144 = 169

= 169

因式分解数字:

169 = 1 3^{2}

= 1 3^{2}

使用根式运算法则:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 5 ) \pm 13}{2 ( - 6 )}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 13}{2 ( - 6 )}, u_{2} = \frac{- ( - 5 ) - 13}{2 ( - 6 )}

u = \frac{- ( - 5 ) + 13}{2 ( - 6 )} : - \frac{3}{2}

\frac{- ( - 5 ) + 13}{2 ( - 6 )}

去除括号:

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

= \frac{5 + 13}{- 2 \cdot 6}

数字相加:

5 + 13 = 18

= \frac{18}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{18}{- 12}

使用分式法则:

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

= - \frac{18}{12}

约分:

6

= - \frac{3}{2}

u = \frac{- ( - 5 ) - 13}{2 ( - 6 )} : \frac{2}{3}

\frac{- ( - 5 ) - 13}{2 ( - 6 )}

去除括号:

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

= \frac{5 - 13}{- 2 \cdot 6}

数字相减:

5 - 13 = - 8

= \frac{- 8}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{- 8}{- 12}

使用分式法则:

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

= \frac{8}{12}

约分:

4

= \frac{2}{3}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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

无解

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

- 1 \leq cos (x) \leq 1

无解

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

cos (x) = \frac{2}{3}

使用反三角函数性质

cos (x) = \frac{2}{3}

cos (x) = \frac{2}{3}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

合并所有解

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

以小数形式表示解

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

作图

流行的例子

3csc(x)-6=0,0<= x<= 360

3 csc (x) - 6 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

-sec^2(x)+1=-5sec(x)-5

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

tan^2(x)+4tan(x)+3=0

tan^{2} (x) + 4 tan (x) + 3 = 0

(1+tan(x))^2+(1-tan(x))^2=sec^2(x)

(1 + tan (x))^{2} + (1 - tan (x))^{2} = sec^{2} (x)

tan(x)=-(sqrt(2))/2

tan (x) = - \frac{2}{2}