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受欢迎的三角函数 >

(tan(x))(sin^2(x)-a)(sec(x)-a)=0,0<a<1

解答

(tan (x)) (sin^{2} (x) - a) (sec (x) - a) = 0, 0 < a < 1

解答

x \in R 无解

求解步骤

(tan (x)) (sin^{2} (x) - a) (sec (x) - a) = 0, 0 < a < 1

分别求解每个部分

tan (x) = 0 or sin^{2} (x) - a = 0 or sec (x) - a = 0

tan (x) = 0, 0 < a < 1 :

无解

tan (x) = 0, 0 < a < 1

tan (x) = 0

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

在

0 < a < 1

范围内的解

无解

sin^{2} (x) - a = 0, 0 < a < 1 :

无解

sin^{2} (x) - a = 0, 0 < a < 1

用替代法求解

sin^{2} (x) - a = 0

令

: sin (x) = u

u^{2} - a = 0

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

u^{2} - a = 0

将

a

到右边

u^{2} - a = 0

两边加上

a

u^{2} - a + a = 0 + a

化简

u^{2} = a

u^{2} = a

对于

x^{2} = f (a)

解为

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

u = a, u = - a

u = sin (x)

代回

sin (x) = a, sin (x) = - a

sin (x) = a, sin (x) = - a

sin (x) = a, 0 < a < 1 :

无解

sin (x) = a, 0 < a < 1

使用反三角函数性质

sin (x) = a

sin (x) = a

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

在

0 < a < 1

范围内的解

无解

sin (x) = - a, 0 < a < 1 :

无解

sin (x) = - a, 0 < a < 1

使用反三角函数性质

sin (x) = - a

sin (x) = - a

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

在

0 < a < 1

范围内的解

无解

合并所有解

无解

sec (x) - a = 0, 0 < a < 1 :

无解

sec (x) - a = 0, 0 < a < 1

将

a

到右边

sec (x) - a = 0

两边加上

a

sec (x) - a + a = 0 + a

化简

sec (x) = a

sec (x) = a

使用反三角函数性质

sec (x) = a

sec (x) = a

的通解

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

在

0 < a < 1

范围内的解

无解

合并所有解

x \in R 无解

作图

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