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受欢迎的三角函数 >

2cos(x)+2sqrt(2)=3sec(x)

解答

2 cos (x) + 22 = 3 sec (x)

解答

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

2 cos (x) + 22 = 3 sec (x)

两边减去

3 sec (x)

2 cos (x) + 22 - 3 sec (x) = 0

使用三角恒等式改写

2 cos (x) + 22 - 3 sec (x)

使用基本三角恒等式:

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

= 2 \cdot \frac{1}{sec ( x )} + 22 - 3 sec (x)

2 \cdot \frac{1}{sec ( x )} = \frac{2}{sec ( x )}

2 \cdot \frac{1}{sec ( x )}

分式相乘:

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

= \frac{1 \cdot 2}{sec ( x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{sec ( x )}

= \frac{2}{sec ( x )} + 22 - 3 sec (x)

\frac{2}{sec ( x )} + 22 - 3 sec (x) = 0

用替代法求解

\frac{2}{sec ( x )} + 22 - 3 sec (x) = 0

令

: sec (x) = u

\frac{2}{u} + 22 - 3 u = 0

\frac{2}{u} + 22 - 3 u = 0 : u = - \frac{2}{3}, u = 2

\frac{2}{u} + 22 - 3 u = 0

在两边乘以

u

\frac{2}{u} + 22 - 3 u = 0

在两边乘以

u

\frac{2}{u} u + 22 u - 3 uu = 0 \cdot u

化简

\frac{2}{u} u + 22 u - 3 uu = 0 \cdot u

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

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

2 + 22 u - 3 u^{2} = 0

2 + 22 u - 3 u^{2} = 0

2 + 22 u - 3 u^{2} = 0

解

2 + 22 u - 3 u^{2} = 0 : u = - \frac{2}{3}, u = 2

2 + 22 u - 3 u^{2} = 0

改写成标准形式

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

- 3 u^{2} + 22 u + 2 = 0

使用求根公式求解

- 3 u^{2} + 22 u + 2 = 0

二次方程求根公式:

若

a = - 3, b = 22, c = 2

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

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

(22)^{2} - 4 (- 3) \cdot 2 = 42

(22)^{2} - 4 (- 3) \cdot 2

使用法则

- (- a) = a

= (22)^{2} + 4 \cdot 3 \cdot 2

(22)^{2} = 2^{3}

(22)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (2)^{2}

(2)^{2} : 2

使用根式运算法则:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2

= 2^{2} \cdot 2

使用指数法则:

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

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

数字相加:

2 + 1 = 3

= 2^{3}

4 \cdot 3 \cdot 2 = 24

4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 24

= 2^{3} + 24

2^{3} = 8

= 8 + 24

数字相加:

8 + 24 = 32

= 32

32

质因数分解:

2^{5}

32

32

除以

232 = 16 \cdot 2

= 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

使用指数法则:

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

= 2^{4} \cdot 2

使用根式运算法则:

= 2 2^{4}

使用根式运算法则:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

整理后得

= 42

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

将解分隔开

u_{1} = \frac{- 2 2 + 4 2}{2 ( - 3 )}, u_{2} = \frac{- 2 2 - 4 2}{2 ( - 3 )}

u = \frac{- 2 2 + 4 2}{2 ( - 3 )} : - \frac{2}{3}

\frac{- 2 2 + 4 2}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 2 2 + 4 2}{- 2 \cdot 3}

同类项相加:

- 22 + 42 = 22

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

数字相乘:

2 \cdot 3 = 6

= \frac{2 2}{- 6}

使用分式法则:

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

= - \frac{2 2}{6}

约分:

2

= - \frac{2}{3}

u = \frac{- 2 2 - 4 2}{2 ( - 3 )} : 2

\frac{- 2 2 - 4 2}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 2 2 - 4 2}{- 2 \cdot 3}

同类项相加:

- 22 - 42 = - 62

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

数字相乘:

2 \cdot 3 = 6

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

使用分式法则:

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

= \frac{6 2}{6}

数字相除:

\frac{6}{6} = 1

= 2

二次方程组的解是:

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

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

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{2}{u} + 22 - 3 u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

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

u = sec (x)

代回

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

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

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

无解

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

sec (x) \leq - 1 or sec (x) \geq 1

无解

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = 2

sec (x) = 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

cot^2(x)-3cot(x)-2=0

sin(2x-23)=-(sqrt(2))/2

cos(3x)+sin(x)=0

sin(x)cos(x)cos(2x)= 1/8