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受欢迎的三角函数 >

3/2 sec(θ)+3cos(θ)=3

解答

\frac{3}{2} sec (θ) + 3 cos (θ) = 3

解答

θ \in R 无解

求解步骤

\frac{3}{2} sec (θ) + 3 cos (θ) = 3

两边减去

3

\frac{3}{2} sec (θ) + 3 cos (θ) - 3 = 0

化简

\frac{3}{2} sec (θ) + 3 cos (θ) - 3 : \frac{3 sec ( θ ) + 6 cos ( θ ) - 6}{2}

\frac{3}{2} sec (θ) + 3 cos (θ) - 3

乘

\frac{3}{2} sec (θ) : \frac{3 sec ( θ )}{2}

\frac{3}{2} sec (θ)

分式相乘:

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

= \frac{3 sec ( θ )}{2}

= \frac{3 sec ( θ )}{2} + 3 cos (θ) - 3

将项转换为分式:

3 cos (θ) = \frac{3 cos ( θ ) 2}{2}, 3 = \frac{3 \cdot 2}{2}

= \frac{3 sec ( θ )}{2} + \frac{3 cos ( θ ) \cdot 2}{2} - \frac{3 \cdot 2}{2}

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{3 sec ( θ ) + 3 cos ( θ ) \cdot 2 - 3 \cdot 2}{2}

数字相乘:

3 \cdot 2 = 6

= \frac{3 sec ( θ ) + 6 cos ( θ ) - 6}{2}

\frac{3 sec ( θ ) + 6 cos ( θ ) - 6}{2} = 0

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

3 sec (θ) + 6 cos (θ) - 6 = 0

使用三角恒等式改写

- 6 + 3 sec (θ) + 6 cos (θ)

使用基本三角恒等式:

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

= - 6 + 3 sec (θ) + 6 \cdot \frac{1}{sec ( θ )}

6 \cdot \frac{1}{sec ( θ )} = \frac{6}{sec ( θ )}

6 \cdot \frac{1}{sec ( θ )}

分式相乘:

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

= \frac{1 \cdot 6}{sec ( θ )}

数字相乘:

1 \cdot 6 = 6

= \frac{6}{sec ( θ )}

= - 6 + 3 sec (θ) + \frac{6}{sec ( θ )}

- 6 + \frac{6}{sec ( θ )} + 3 sec (θ) = 0

用替代法求解

- 6 + \frac{6}{sec ( θ )} + 3 sec (θ) = 0

令

: sec (θ) = u

- 6 + \frac{6}{u} + 3 u = 0

- 6 + \frac{6}{u} + 3 u = 0 : u = 1 + i, u = 1 - i

- 6 + \frac{6}{u} + 3 u = 0

在两边乘以

u

- 6 + \frac{6}{u} + 3 u = 0

在两边乘以

u

- 6 u + \frac{6}{u} u + 3 uu = 0 \cdot u

化简

- 6 u + \frac{6}{u} u + 3 uu = 0 \cdot u

化简

\frac{6}{u} u : 6

\frac{6}{u} u

分式相乘:

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

= \frac{6 u}{u}

约分:

u

= 6

化简

3 uu : 3 u^{2}

3 uu

使用指数法则:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 6 u + 6 + 3 u^{2} = 0

- 6 u + 6 + 3 u^{2} = 0

- 6 u + 6 + 3 u^{2} = 0

解

- 6 u + 6 + 3 u^{2} = 0 : u = 1 + i, u = 1 - i

- 6 u + 6 + 3 u^{2} = 0

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 3, b = - 6, c = 6

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 3 \cdot 6}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 3 \cdot 6}{2 \cdot 3}

化简

(- 6)^{2} - 4 \cdot 3 \cdot 6 : 6 i

(- 6)^{2} - 4 \cdot 3 \cdot 6

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 3 \cdot 6

数字相乘:

4 \cdot 3 \cdot 6 = 72

= 6^{2} - 72

使用虚数运算法则:

- a = i a

= i 72 - 6^{2}

- 6^{2} + 72 = 6

- 6^{2} + 72

6^{2} = 36

= - 36 + 72

数字相加/相减:

- 36 + 72 = 36

= 36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

6^{2} = 6

= 6

= 6 i

u_{1, 2} = \frac{- ( - 6 ) \pm 6 i}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 6 ) + 6 i}{2 \cdot 3}, u_{2} = \frac{- ( - 6 ) - 6 i}{2 \cdot 3}

u = \frac{- ( - 6 ) + 6 i}{2 \cdot 3} : 1 + i

\frac{- ( - 6 ) + 6 i}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{6 + 6 i}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{6 + 6 i}{6}

分解

6 + 6 i : 6 (1 + i)

6 + 6 i

改写为

= 6 \cdot 1 + 6 i

因式分解出通项

6

= 6 (1 + i)

= \frac{6 ( 1 + i )}{6}

数字相除:

\frac{6}{6} = 1

= 1 + i

u = \frac{- ( - 6 ) - 6 i}{2 \cdot 3} : 1 - i

\frac{- ( - 6 ) - 6 i}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{6 - 6 i}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{6 - 6 i}{6}

分解

6 - 6 i : 6 (1 - i)

6 - 6 i

改写为

= 6 \cdot 1 - 6 i

因式分解出通项

6

= 6 (1 - i)

= \frac{6 ( 1 - i )}{6}

数字相除:

\frac{6}{6} = 1

= 1 - i

二次方程组的解是:

u = 1 + i, u = 1 - i

u = 1 + i, u = 1 - i

u = sec (θ)

代回

sec (θ) = 1 + i, sec (θ) = 1 - i

sec (θ) = 1 + i, sec (θ) = 1 - i

sec (θ) = 1 + i :

无解

sec (θ) = 1 + i

无解

sec (θ) = 1 - i :

无解

sec (θ) = 1 - i

无解

合并所有解

θ \in R 无解

作图

流行的例子

12cos^2(x)+2cos(x)-2=0

sin(θ)=0,0<= θ<= 2pi