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受欢迎的三角函数 >

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

解答

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

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

+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

求解步骤

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

两边减去

\frac{2}{3}

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

化简

sec^{2} (x) - 2 tan^{2} (x) - \frac{2}{3} : \frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3}

sec^{2} (x) - 2 tan^{2} (x) - \frac{2}{3}

将项转换为分式:

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

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

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{sec ^{2} ( x ) \cdot 3 - 2 tan ^{2} ( x ) \cdot 3 - 2}{3}

数字相乘:

2 \cdot 3 = 6

= \frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3}

\frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3} = 0

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

3 sec^{2} (x) - 6 tan^{2} (x) - 2 = 0

使用三角恒等式改写

- 2 + 3 sec^{2} (x) - 6 tan^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1)

化简

- 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1) : - 3 sec^{2} (x) + 4

- 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1)

乘开

- 6 (sec^{2} (x) - 1) : - 6 sec^{2} (x) + 6

- 6 (sec^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = - 6, b = sec^{2} (x), c = 1

= - 6 sec^{2} (x) - (- 6) \cdot 1

使用加减运算法则

- (- a) = a

= - 6 sec^{2} (x) + 6 \cdot 1

数字相乘:

6 \cdot 1 = 6

= - 6 sec^{2} (x) + 6

= - 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6

化简

- 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6 : - 3 sec^{2} (x) + 4

- 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6

同类项相加:

3 sec^{2} (x) - 6 sec^{2} (x) = - 3 sec^{2} (x)

= - 2 - 3 sec^{2} (x) + 6

对同类项分组

= - 3 sec^{2} (x) - 2 + 6

数字相加/相减:

- 2 + 6 = 4

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

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

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

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

用替代法求解

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

令

: sec (x) = u

4 - 3 u^{2} = 0

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

4 - 3 u^{2} = 0

将

4

到右边

4 - 3 u^{2} = 0

两边减去

4

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

化简

- 3 u^{2} = - 4

- 3 u^{2} = - 4

两边除以

- 3

- 3 u^{2} = - 4

两边除以

- 3

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

化简

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

u = sec (x)

代回

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

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

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

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

作图

流行的例子

cos^2(x)-3cos(x)-1=0

cos^{2} (x) - 3 cos (x) - 1 = 0

cos(3x)=-sin(x+pi/6)

cos (3 x) = - sin (x + \frac{π}{6})

sin (θ) = \frac{7}{10}

tan (5 x) = 1

sec (5 x) = 2