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受欢迎的三角函数 >

sec^2(x)=3tan^2(x)-1

解答

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

解答

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 tan^{2} (x) - 1

sec^{2} (x) - 3 tan^{2} (x) + 1 = 0

使用三角恒等式改写

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - 3, b = sec^{2} (x), c = 1

= - 3 sec^{2} (x) - (- 3) \cdot 1

使用加减运算法则

- (- a) = a

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

数字相乘:

3 \cdot 1 = 3

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

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

化简

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

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

同类项相加:

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

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

对同类项分组

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

数字相加:

1 + 3 = 4

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

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

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

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

用替代法求解

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

令

: sec (x) = u

4 - 2 u^{2} = 0

4 - 2 u^{2} = 0 : u = 2, u = - 2

4 - 2 u^{2} = 0

将

4

到右边

4 - 2 u^{2} = 0

两边减去

4

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

化简

- 2 u^{2} = - 4

- 2 u^{2} = - 4

两边除以

- 2

- 2 u^{2} = - 4

两边除以

- 2

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

化简

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

化简

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

\frac{- 2 u ^{2}}{- 2}

使用分式法则:

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

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

数字相除:

\frac{2}{2} = 1

= u^{2}

化简

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

使用分式法则:

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

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

u^{2} = 2

u^{2} = 2

u^{2} = 2

对于

x^{2} = f (a)

解为

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

u = 2, u = - 2

u = sec (x)

代回

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

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

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 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{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

作图

流行的例子

csc(x)+cot(x)=(sqrt(3))/3

sin(θ)+1=cos(θ)

sqrt(2)sin^2(θ)-sin(θ)=0

2sin^2(x)=2-sqrt(3)cos(x)

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