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受欢迎的三角函数 >

tan^2(x)-4sec(x)+5=0

解答

tan^{2} (x) - 4 sec (x) + 5 = 0

解答

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

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

tan^{2} (x) - 4 sec (x) + 5 = 0

使用三角恒等式改写

5 + tan^{2} (x) - 4 sec (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

= 5 + sec^{2} (x) - 1 - 4 sec (x)

化简

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

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

对同类项分组

= sec^{2} (x) - 4 sec (x) + 5 - 1

数字相加/相减:

5 - 1 = 4

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

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

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

用替代法求解

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

令

: sec (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 4, c = 4

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

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

(- 4)^{2} - 4 \cdot 1 \cdot 4 = 0

(- 4)^{2} - 4 \cdot 1 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 4)^{2} = 4^{2}

= 4^{2} - 4 \cdot 1 \cdot 4

数字相乘:

4 \cdot 1 \cdot 4 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

数字相减:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 1}

u = \frac{- ( - 4 )}{2 \cdot 1}

\frac{- ( - 4 )}{2 \cdot 1} = 2

\frac{- ( - 4 )}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{4}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

u = 2

二次方程组的解是:

u = 2

u = sec (x)

代回

sec (x) = 2

sec (x) = 2

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

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

合并所有解

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

作图

流行的例子

cot (θ) = \frac{3}{4}

tan (x) = \frac{7}{3}

sqrt(3)=sqrt(3)cos(2θ)+sin(2θ)

3 = 3 cos (2 θ) + sin (2 θ)

6cos(x)+6sin(x)=3sqrt(6)

6 cos (x) + 6 sin (x) = 36

3 sin (x) + 1 = 0