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受欢迎的三角函数 >

sec^2(2x)-2tan(2x)=0

解答

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

解答

x = \frac{π}{8} + \frac{πn}{2}

+1

度数

x = 22. 5^{\circ} + 9 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

用替代法求解

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

令

: tan (2 x) = u

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

1 + u^{2} - 2 u = 0 : u = 1

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

改写成标准形式

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

u^{2} - 2 u + 1 = 0

使用求根公式求解

u^{2} - 2 u + 1 = 0

二次方程求根公式:

若

a = 1, b = - 2, c = 1

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

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

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

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 2^{2} - 4

2^{2} = 4

= 4 - 4

数字相减:

4 - 4 = 0

= 0

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

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

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

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

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

u = 1

二次方程组的解是:

u = 1

u = tan (2 x)

代回

tan (2 x) = 1

tan (2 x) = 1

tan (2 x) = 1 : x = \frac{π}{8} + \frac{πn}{2}

tan (2 x) = 1

tan (2 x) = 1

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}

2 x = \frac{π}{4} + πn

2 x = \frac{π}{4} + πn

解

2 x = \frac{π}{4} + πn : x = \frac{π}{8} + \frac{πn}{2}

2 x = \frac{π}{4} + πn

两边除以

2

2 x = \frac{π}{4} + πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{4}}{2} + \frac{πn}{2} : \frac{π}{8} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

使用分式法则:

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

= \frac{π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{π}{8}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

合并所有解

x = \frac{π}{8} + \frac{πn}{2}

作图

流行的例子

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6sin^2(x)+5cos(x)=7

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tan(6x)-3tan(3x)=0

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