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

解答

x = \frac{π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

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

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

化简

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

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

将分母和分子乘以

sin (x)

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

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

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

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

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

化简

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

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

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

将项转换为分式:

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

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

乘开

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

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

同类项相加:

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

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

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

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

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

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

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

分解

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

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

使用指数法则:

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

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

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

因式分解出通项

sin (x)

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

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

分别求解每个部分

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

sin (x) - cos (x) = 0 : x = \frac{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

两边加上

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{π}{4} + πn

x = \frac{π}{4} + πn

合并所有解

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

因为方程对以下值无定义:

2 πn, π + 2 πn

x = \frac{π}{4} + πn

作图

流行的例子

3tan(x+45)=sqrt(3)

3 tan (x + 4 5^{\circ}) = 3

0 = sin (π x)

sec(θ)csc(θ)=1

sec (θ) csc (θ) = 1

证明 cos(a+270)=sin(a)

p ro v e cos (a + 27 0^{\circ}) = sin (a)

1+sin(θ)=2-sin(θ)

1 + sin (θ) = 2 - sin (θ)