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受欢迎的三角函数 >

2sec^2(x)-3tan(x)=11

解答

2 sec^{2} (x) - 3 tan (x) = 11

解答

x = 1.24904 \dots + πn, x = - 0.98279 \dots + πn

+1

度数

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 56.30993 \dots^{\circ} + 18 0^{\circ} n

求解步骤

2 sec^{2} (x) - 3 tan (x) = 11

两边减去

11

2 sec^{2} (x) - 3 tan (x) - 11 = 0

使用三角恒等式改写

- 11 + 2 sec^{2} (x) - 3 tan (x)

使用毕达哥拉斯恒等式:

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

= - 11 + 2 (tan^{2} (x) + 1) - 3 tan (x)

化简

- 11 + 2 (tan^{2} (x) + 1) - 3 tan (x) : 2 tan^{2} (x) - 3 tan (x) - 9

- 11 + 2 (tan^{2} (x) + 1) - 3 tan (x)

乘开

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

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

使用分配律:

a (b + c) = ab + a c

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

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

数字相乘:

2 \cdot 1 = 2

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

= - 11 + 2 tan^{2} (x) + 2 - 3 tan (x)

化简

- 11 + 2 tan^{2} (x) + 2 - 3 tan (x) : 2 tan^{2} (x) - 3 tan (x) - 9

- 11 + 2 tan^{2} (x) + 2 - 3 tan (x)

对同类项分组

= 2 tan^{2} (x) - 3 tan (x) - 11 + 2

数字相加/相减:

- 11 + 2 = - 9

= 2 tan^{2} (x) - 3 tan (x) - 9

= 2 tan^{2} (x) - 3 tan (x) - 9

= 2 tan^{2} (x) - 3 tan (x) - 9

- 9 + 2 tan^{2} (x) - 3 tan (x) = 0

用替代法求解

- 9 + 2 tan^{2} (x) - 3 tan (x) = 0

令

: tan (x) = u

- 9 + 2 u^{2} - 3 u = 0

- 9 + 2 u^{2} - 3 u = 0 : u = 3, u = - \frac{3}{2}

- 9 + 2 u^{2} - 3 u = 0

改写成标准形式

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

2 u^{2} - 3 u - 9 = 0

使用求根公式求解

2 u^{2} - 3 u - 9 = 0

二次方程求根公式:

若

a = 2, b = - 3, c = - 9

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 9 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 9 )}{2 \cdot 2}

(- 3)^{2} - 4 \cdot 2 (- 9) = 9

(- 3)^{2} - 4 \cdot 2 (- 9)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 9

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 2 \cdot 9

数字相乘:

4 \cdot 2 \cdot 9 = 72

= 3^{2} + 72

3^{2} = 9

= 9 + 72

数字相加:

9 + 72 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- ( - 3 ) \pm 9}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 9}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 9}{2 \cdot 2}

u = \frac{- ( - 3 ) + 9}{2 \cdot 2} : 3

\frac{- ( - 3 ) + 9}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 + 9}{2 \cdot 2}

数字相加:

3 + 9 = 12

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

数字相乘:

2 \cdot 2 = 4

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 3 ) - 9}{2 \cdot 2} : - \frac{3}{2}

\frac{- ( - 3 ) - 9}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 - 9}{2 \cdot 2}

数字相减:

3 - 9 = - 6

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 6}{4}

使用分式法则:

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

= - \frac{6}{4}

约分:

2

= - \frac{3}{2}

二次方程组的解是:

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

u = tan (x)

代回

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

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

tan (x) = 3 : x = arctan (3) + πn

tan (x) = 3

使用反三角函数性质

tan (x) = 3

tan (x) = 3

的通解

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

x = arctan (3) + πn

x = arctan (3) + πn

tan (x) = - \frac{3}{2} : x = arctan (- \frac{3}{2}) + πn

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

使用反三角函数性质

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

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

的通解

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

合并所有解

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

以小数形式表示解

x = 1.24904 \dots + πn, x = - 0.98279 \dots + πn

作图

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