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 >

3sec^2(x)-5tan(x)=3

解答

3 sec^{2} (x) - 5 tan (x) = 3

解答

x = 1.03037 \dots + πn, x = πn

+1

度数

x = 59.03624 \dots^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n

求解步骤

3 sec^{2} (x) - 5 tan (x) = 3

两边减去

3

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

使用三角恒等式改写

- 3 + 3 sec^{2} (x) - 5 tan (x)

使用毕达哥拉斯恒等式:

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

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

化简

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

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

乘开

3 (tan^{2} (x) + 1) : 3 tan^{2} (x) + 3

3 (tan^{2} (x) + 1)

使用分配律:

a (b + c) = ab + a c

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

= 3 tan^{2} (x) + 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

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

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

化简

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

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

对同类项分组

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

- 3 + 3 = 0

= 3 tan^{2} (x) - 5 tan (x)

= 3 tan^{2} (x) - 5 tan (x)

= 3 tan^{2} (x) - 5 tan (x)

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

用替代法求解

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

令

: tan (x) = u

3 u^{2} - 5 u = 0

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

3 u^{2} - 5 u = 0

使用求根公式求解

3 u^{2} - 5 u = 0

二次方程求根公式:

若

a = 3, b = - 5, c = 0

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

(- 5)^{2} - 4 \cdot 3 \cdot 0 = 5

(- 5)^{2} - 4 \cdot 3 \cdot 0

使用指数法则:

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

若

n

是偶数

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

= 5^{2} - 4 \cdot 3 \cdot 0

使用法则

0 \cdot a = 0

= 5^{2} - 0

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

= 5^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 5

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

将解分隔开

u_{1} = \frac{- ( - 5 ) + 5}{2 \cdot 3}, u_{2} = \frac{- ( - 5 ) - 5}{2 \cdot 3}

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

\frac{- ( - 5 ) + 5}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{5 + 5}{2 \cdot 3}

数字相加:

5 + 5 = 10

= \frac{10}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{10}{6}

约分:

2

= \frac{5}{3}

u = \frac{- ( - 5 ) - 5}{2 \cdot 3} : 0

\frac{- ( - 5 ) - 5}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{5 - 5}{2 \cdot 3}

数字相减:

5 - 5 = 0

= \frac{0}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{0}{6}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

二次方程组的解是:

u = \frac{5}{3}, u = 0

u = tan (x)

代回

tan (x) = \frac{5}{3}, tan (x) = 0

tan (x) = \frac{5}{3}, tan (x) = 0

tan (x) = \frac{5}{3} : x = arctan (\frac{5}{3}) + πn

tan (x) = \frac{5}{3}

使用反三角函数性质

tan (x) = \frac{5}{3}

tan (x) = \frac{5}{3}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{5}{3}) + πn

x = arctan (\frac{5}{3}) + πn

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

合并所有解

x = arctan (\frac{5}{3}) + πn, x = πn

以小数形式表示解

x = 1.03037 \dots + πn, x = πn

作图

流行的例子

sin(2x)cos(6x)-cos(2x)sin(6x)=-0.55

sin (2 x) cos (6 x) - cos (2 x) sin (6 x) = - 0.55

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

3tan^3(θ)-tan(θ)=3tan^2(θ)-1

3 tan^{3} (θ) - tan (θ) = 3 tan^{2} (θ) - 1

sin (θ) = - 0.7

sin (θ) = - 0.1