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受欢迎的三角函数 >

(sin(x)+cos(x))/(sin(x))=1 1/(tan(x))

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

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

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

化简

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

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

sin (x), tan (x)

的最小公倍数:

sin (x) tan (x)

sin (x), tan (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

tan (x)

中的因子组成的表达式

= sin (x) tan (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (x) tan (x)

对于

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

将分母和分子乘以

tan (x)

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

对于

\frac{1}{tan ( x )} :

将分母和分子乘以

sin (x)

\frac{1}{tan ( x )} = \frac{1 \cdot sin ( x )}{tan ( x ) sin ( x )} = \frac{sin ( x )}{sin ( x ) tan ( x )}

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

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

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

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

使用三角恒等式改写

- sin (x) + (cos (x) + sin (x)) tan (x)

使用基本三角恒等式:

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

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

化简

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

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

乘

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

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

分式相乘:

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

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

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

将项转换为分式:

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

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

乘开

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

sin (x) (cos (x) + sin (x)) - sin (x) cos (x)

乘开

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

sin (x) (cos (x) + sin (x))

使用分配律:

a (b + c) = ab + a c

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

= sin (x) cos (x) + sin (x) sin (x)

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

同类项相加:

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

= sin^{2} (x)

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

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

\frac{sin ^{2} ( x )}{cos ( x )} = 0

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

sin^{2} (x) = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

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

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

arctan(x)=(-pi)/2

arctan (x) = \frac{- π}{2}

sin(4x-3)=cos(5x-6)

sin (4 x - 3) = cos (5 x - 6)

6sin(2x-30)+2=0

6 sin (2 x - 3 0^{\circ}) + 2 = 0

sin(x)=sin(pi/(10))

sin (x) = sin (\frac{π}{10})

sin(3x)-sin(x)=-2sin(2x)sin(x)

sin (3 x) - sin (x) = - 2 sin (2 x) sin (x)