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)+4cos^2(x)=7

解答

3 sec^{2} (x) + 4 cos^{2} (x) = 7

解答

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

3 sec^{2} (x) + 4 cos^{2} (x) = 7

两边减去

7

3 sec^{2} (x) + 4 cos^{2} (x) - 7 = 0

使用三角恒等式改写

- 7 + 3 sec^{2} (x) + 4 cos^{2} (x)

使用基本三角恒等式:

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

= - 7 + 3 sec^{2} (x) + 4 (\frac{1}{sec ( x )})^{2}

4 (\frac{1}{sec ( x )})^{2} = \frac{4}{sec ^{2} ( x )}

4 (\frac{1}{sec ( x )})^{2}

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

= 4 \cdot \frac{1}{sec ^{2} ( x )}

分式相乘:

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

= \frac{1 \cdot 4}{sec ^{2} ( x )}

数字相乘:

1 \cdot 4 = 4

= \frac{4}{sec ^{2} ( x )}

= - 7 + 3 sec^{2} (x) + \frac{4}{sec ^{2} ( x )}

- 7 + \frac{4}{sec ^{2} ( x )} + 3 sec^{2} (x) = 0

用替代法求解

- 7 + \frac{4}{sec ^{2} ( x )} + 3 sec^{2} (x) = 0

令

: sec (x) = u

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

在两边乘以

u^{2}

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

在两边乘以

u^{2}

- 7 u^{2} + \frac{4}{u ^{2}} u^{2} + 3 u^{2} u^{2} = 0 \cdot u^{2}

化简

- 7 u^{2} + \frac{4}{u ^{2}} u^{2} + 3 u^{2} u^{2} = 0 \cdot u^{2}

化简

\frac{4}{u ^{2}} u^{2} : 4

\frac{4}{u ^{2}} u^{2}

分式相乘:

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

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

约分:

u^{2}

= 4

化简

3 u^{2} u^{2} : 3 u^{4}

3 u^{2} u^{2}

使用指数法则:

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

u^{2} u^{2} = u^{2 + 2}

= 3 u^{2 + 2}

数字相加:

2 + 2 = 4

= 3 u^{4}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

- 7 u^{2} + 4 + 3 u^{4} = 0

- 7 u^{2} + 4 + 3 u^{4} = 0

- 7 u^{2} + 4 + 3 u^{4} = 0

解

- 7 u^{2} + 4 + 3 u^{4} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

- 7 u^{2} + 4 + 3 u^{4} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

3 u^{4} - 7 u^{2} + 4 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

3 v^{2} - 7 v + 4 = 0

解

3 v^{2} - 7 v + 4 = 0 : v = \frac{4}{3}, v = 1

3 v^{2} - 7 v + 4 = 0

使用求根公式求解

3 v^{2} - 7 v + 4 = 0

二次方程求根公式:

若

a = 3, b = - 7, c = 4

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

v_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

将解分隔开

v_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, v_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

v = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

数字相加:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{8}{6}

约分:

2

= \frac{4}{3}

v = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

数字相减:

7 - 1 = 6

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

数字相乘:

2 \cdot 3 = 6

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

v = \frac{4}{3}, v = 1

v = \frac{4}{3}, v = 1

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{4}{3} : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u^{2} = \frac{4}{3}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{4}{3}, u = - \frac{4}{3}

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

\frac{4}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2}{3}

\frac{2}{3}

有理化:

\frac{2 3}{3}

\frac{2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

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

- \frac{4}{3}

化简

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

- \frac{2}{3}

有理化:

- \frac{2 3}{3}

- \frac{2}{3}

乘以共轭根式

\frac{3}{3}

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

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

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

解

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

解为

u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

验证解

找到无定义的点（奇点）:

u = 0

取

- 7 + \frac{4}{u ^{2}} + 3 u^{2}

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

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}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

u = sec (x)

代回

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}, sec (x) = 1, sec (x) = - 1

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}, sec (x) = 1, sec (x) = - 1

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

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

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

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

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

sec (x) = 1 : x = 2 πn

sec (x) = 1

sec (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

sec (x) = - 1

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 = π + 2 πn

x = π + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

作图

流行的例子

cos (θ) = \frac{43}{90}

8 cos (2 x - 5) = 3

(sqrt(116))/(sin(90))= 4/(sin(x))

\frac{116}{sin ( 9 0 ^{\circ} )} = \frac{4}{sin ( x )}

3/(cos^2(x))=(7+4)/(cot(x))

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

sin (x) = 0.01