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受欢迎的三角函数 >

sec(2x)+tan(2x)=2

解答

sec (2 x) + tan (2 x) = 2

解答

x = 0.32175 \dots + πn

+1

度数

x = 18.43494 \dots^{\circ} + 18 0^{\circ} n

求解步骤

sec (2 x) + tan (2 x) = 2

两边减去

2

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

用 sin, cos 表示

- 2 + sec (2 x) + tan (2 x)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

= - 2 + \frac{1}{cos ( 2 x )} + tan (2 x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 2 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )}

化简

- 2 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )} : \frac{- 2 cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

- 2 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )}

合并分式

\frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )} : \frac{1 + sin ( 2 x )}{cos ( 2 x )}

使用法则

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

= \frac{1 + sin ( 2 x )}{cos ( 2 x )}

= - 2 + \frac{sin ( 2 x ) + 1}{cos ( 2 x )}

将项转换为分式:

2 = \frac{2 cos ( 2 x )}{cos ( 2 x )}

= - \frac{2 cos ( 2 x )}{cos ( 2 x )} + \frac{1 + sin ( 2 x )}{cos ( 2 x )}

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{- 2 cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

= \frac{- 2 cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

\frac{1 + sin ( 2 x ) - 2 cos ( 2 x )}{cos ( 2 x )} = 0

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

1 + sin (2 x) - 2 cos (2 x) = 0

使用三角恒等式改写

1 + sin (2 x) - 2 cos (2 x)

使用倍角公式:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

= 1 + sin (2 x) - 2 (cos^{2} (x) - sin^{2} (x))

1 + sin (2 x) = (sin (x) + cos (x))^{2}

1 + sin (2 x)

使用毕达哥拉斯恒等式:

1 = cos^{2} (x) + sin^{2} (x)

= (cos^{2} (x) + sin^{2} (x)) + sin (2 x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= cos^{2} (x) + sin^{2} (x) + 2 sin (x) cos (x)

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) = (sin (x) + cos (x))^{2}

= (sin (x) + cos (x))^{2}

= (sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x))

化简

(sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x)) : 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

(sin (x) + cos (x))^{2} - 2 (cos^{2} (x) - sin^{2} (x))

(sin (x) + cos (x))^{2} : sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 (cos^{2} (x) - sin^{2} (x))

乘开

- 2 (cos^{2} (x) - sin^{2} (x)) : - 2 cos^{2} (x) + 2 sin^{2} (x)

- 2 (cos^{2} (x) - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 2, b = cos^{2} (x), c = sin^{2} (x)

= - 2 cos^{2} (x) - (- 2) sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 2 cos^{2} (x) + 2 sin^{2} (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x)

化简

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x) : 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x) - 2 cos^{2} (x) + 2 sin^{2} (x)

同类项相加:

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

= sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x) + 2 sin^{2} (x)

同类项相加:

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

= 3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

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

分解

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x) : (3 sin (x) - cos (x)) (sin (x) + cos (x))

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x)

将表达式拆分成组

3 sin^{2} (x) + 2 sin (x) cos (x) - cos^{2} (x)

定义

3

的因数:

1, 3

3

约数 (因数)

找到

3

的质因数:

3

3

3

是质数，因此无法因数分解

= 3

加 1

1

3

的因数

1, 3

3

的负因数:

- 1, - 3

将因数乘以

- 1

得到负因数

- 1, - 3

对于每两个因数

u * v = - 3

，检验是否

u + v = 2

检验

u = 1, v = - 3 : u * v = - 3, u + v = - 2 \Rightarrow

假检验

u = 3, v = - 1 : u * v = - 3, u + v = 2 \Rightarrow

真

u = 3, v = - 1

分组为

(a x^{2} + ux y) + (vx y + c y^{2})

(3 sin^{2} (x) - sin (x) cos (x)) + (3 sin (x) cos (x) - cos^{2} (x))

= (3 sin^{2} (x) - sin (x) cos (x)) + (3 sin (x) cos (x) - cos^{2} (x))

从

3 sin^{2} (x) - sin (x) cos (x)

分解出因式

sin (x)

:

sin (x) (3 sin (x) - cos (x))

3 sin^{2} (x) - sin (x) cos (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= 3 sin (x) sin (x) - sin (x) cos (x)

因式分解出通项

sin (x)

= sin (x) (3 sin (x) - cos (x))

从

3 sin (x) cos (x) - cos^{2} (x)

分解出因式

cos (x)

:

cos (x) (3 sin (x) - cos (x))

3 sin (x) cos (x) - cos^{2} (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

cos^{2} (x) = cos (x) cos (x)

= 3 sin (x) cos (x) - cos (x) cos (x)

因式分解出通项

cos (x)

= cos (x) (3 sin (x) - cos (x))

= sin (x) (3 sin (x) - cos (x)) + cos (x) (3 sin (x) - cos (x))

因式分解出通项

3 sin (x) - cos (x)

= (3 sin (x) - cos (x)) (sin (x) + cos (x))

(3 sin (x) - cos (x)) (sin (x) + cos (x)) = 0

分别求解每个部分

3 sin (x) - cos (x) = 0 or sin (x) + cos (x) = 0

3 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{3}) + πn

3 sin (x) - cos (x) = 0

使用三角恒等式改写

3 sin (x) - cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{3 sin ( x )}{cos ( x )} - 1 = 0

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

将

1

到右边

3 tan (x) - 1 = 0

两边加上

1

3 tan (x) - 1 + 1 = 0 + 1

化简

3 tan (x) = 1

3 tan (x) = 1

两边除以

3

3 tan (x) = 1

两边除以

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

化简

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

使用反三角函数性质

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

sin (x) + cos (x) = 0 : x = \frac{3 π}{4} + πn

sin (x) + cos (x) = 0

使用三角恒等式改写

sin (x) + cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} + 1 = 0

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) + 1 = 0

tan (x) + 1 = 0

将

1

到右边

tan (x) + 1 = 0

两边减去

1

tan (x) + 1 - 1 = 0 - 1

化简

tan (x) = - 1

tan (x) = - 1

tan (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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

合并所有解

x = arctan (\frac{1}{3}) + πn, x = \frac{3 π}{4} + πn

因为方程对以下值无定义:

\frac{3 π}{4} + πn

x = arctan (\frac{1}{3}) + πn

以小数形式表示解

x = 0.32175 \dots + πn

作图

流行的例子

sec(2x)+tan(2x)=9

sec (2 x) + tan (2 x) = 9

tan(4x)-tan(2x)=0

tan (4 x) - tan (2 x) = 0

solvefor t,sin(t)+2sin(2t)=0

so l v e f or t, sin (t) + 2 sin (2 t) = 0

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

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

4sin(x)=2.54648

4 sin (x) = 2.54648