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受欢迎的三角函数 >

(sec(x)sin(x))/(tan(x)cot(x))=sin^2(x)

解答

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

解答

x \in R 无解

求解步骤

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

两边减去

sin^{2} (x)

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) = 0

化简

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) : \frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x)

将项转换为分式:

sin^{2} (x) = \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

= \frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( 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{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )} = 0

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

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

分解

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x) : sin (x) (sec (x) - sin (x) tan (x) cot (x))

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

使用指数法则:

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

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

= sec (x) sin (x) - sin (x) sin (x) tan (x) cot (x)

因式分解出通项

sin (x)

= sin (x) (sec (x) - sin (x) tan (x) cot (x))

sin (x) (sec (x) - sin (x) tan (x) cot (x)) = 0

分别求解每个部分

sin (x) = 0 or sec (x) - sin (x) tan (x) cot (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sec (x) - sin (x) tan (x) cot (x) = 0 :

无解

sec (x) - sin (x) tan (x) cot (x) = 0

用 sin, cos 表示

sec (x) - cot (x) sin (x) tan (x)

使用基本三角恒等式:

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

= \frac{1}{cos ( x )} - cot (x) sin (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

约分:

cos (x)

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

约分:

sin (x)

= sin (x)

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

将项转换为分式:

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

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

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

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

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

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

使用三角恒等式改写

1 - cos (x) sin (x)

使用倍角公式:

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

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

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

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

将

1

到右边

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

两边减去

1

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

化简

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

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

在两边乘以

2

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

在两边乘以

2

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

化简

- sin (2 x) = - 2

- sin (2 x) = - 2

两边除以

- 1

- sin (2 x) = - 2

两边除以

- 1

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

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = 2 πn, x = π + 2 πn

因为方程对以下值无定义:

2 πn, π + 2 πn

x \in R 无解

作图

流行的例子

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

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

sin(x)=-3/4 ,pi<x<(3pi)/4

sin (x) = - \frac{3}{4}, π < x < \frac{3 π}{4}

cos(2x)+cos(x)=0,0<= x<2pi

cos (2 x) + cos (x) = 0, 0 \leq x < 2 π

10 cos^{2} (x) = 5

cos (θ) = \frac{5}{9}