zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

sin (x)

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

- sin (x) - cos (x) \frac{cos ( x )}{sin ( x )} + \frac{1}{cos ( x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{- cos ^{2} ( x ) sin ^{2} ( x ) - cos ^{4} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

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

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

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

分式相乘:

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

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

分式相乘:

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

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

乘以:

1 \cdot sin (x) = sin (x)

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

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

cos (x) cos (x)

使用指数法则:

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

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

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

将项转换为分式:

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

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

1, sin (x), cos^{2} (x)

的最小公倍数:

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

1, sin (x), cos^{2} (x)

最小公倍数 (LCM)

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= cos^{2} (x) sin (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

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

对于

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

将分母和分子乘以

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

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

对于

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

将分母和分子乘以

cos^{2} (x)

\frac{cos ^{2} ( x )}{sin ( x )} = \frac{cos ^{2} ( x ) cos ^{2} ( x )}{sin ( x ) cos ^{2} ( x )} = \frac{cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )}

对于

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

将分母和分子乘以

sin (x)

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

= - \frac{cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )} - \frac{cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( 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{- cos ^{2} ( x ) sin ^{2} ( x ) - cos ^{4} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

= \frac{- cos ^{2} ( x ) sin ^{2} ( x ) - cos ^{4} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )}

\frac{- cos ^{4} ( x ) + sin ^{2} ( x ) - cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )} = 0

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

- cos^{4} (x) + sin^{2} (x) - cos^{2} (x) sin^{2} (x) = 0

使用三角恒等式改写

- cos^{4} (x) + sin^{2} (x) - cos^{2} (x) sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - cos^{4} (x) + sin^{2} (x) - (1 - sin^{2} (x)) sin^{2} (x)

化简

- cos^{4} (x) + sin^{2} (x) - (1 - sin^{2} (x)) sin^{2} (x) : - cos^{4} (x) + sin^{4} (x)

- cos^{4} (x) + sin^{2} (x) - (1 - sin^{2} (x)) sin^{2} (x)

= - cos^{4} (x) + sin^{2} (x) - sin^{2} (x) (1 - sin^{2} (x))

乘开

- sin^{2} (x) (1 - sin^{2} (x)) : - sin^{2} (x) + sin^{4} (x)

- sin^{2} (x) (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - sin^{2} (x), b = 1, c = sin^{2} (x)

= - sin^{2} (x) \cdot 1 - (- sin^{2} (x)) sin^{2} (x)

使用加减运算法则

- (- a) = a

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

化简

- 1 \cdot sin^{2} (x) + sin^{2} (x) sin^{2} (x) : - sin^{2} (x) + sin^{4} (x)

- 1 \cdot sin^{2} (x) + sin^{2} (x) sin^{2} (x)

1 \cdot sin^{2} (x) = sin^{2} (x)

1 \cdot sin^{2} (x)

乘以:

1 \cdot sin^{2} (x) = sin^{2} (x)

= sin^{2} (x)

sin^{2} (x) sin^{2} (x) = sin^{4} (x)

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

使用指数法则:

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

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

= sin^{2 + 2} (x)

数字相加:

2 + 2 = 4

= sin^{4} (x)

= - sin^{2} (x) + sin^{4} (x)

= - sin^{2} (x) + sin^{4} (x)

= - cos^{4} (x) + sin^{2} (x) - sin^{2} (x) + sin^{4} (x)

同类项相加:

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

= - cos^{4} (x) + sin^{4} (x)

= - cos^{4} (x) + sin^{4} (x)

- cos^{4} (x) + sin^{4} (x) = 0

分解

- cos^{4} (x) + sin^{4} (x) : (sin^{2} (x) + cos^{2} (x)) (sin (x) + cos (x)) (sin (x) - cos (x))

- cos^{4} (x) + sin^{4} (x)

将

sin^{4} (x) - cos^{4} (x)

改写为

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

sin^{4} (x) - cos^{4} (x)

使用指数法则:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

= (sin^{2} (x))^{2} - cos^{4} (x)

使用指数法则:

a^{b c} = (a^{b})^{c}

cos^{4} (x) = (cos^{2} (x))^{2}

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

分解

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= (- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

化简

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1 : (- cos (x) + sin (x)) (cos (x) + sin (x))

(- cos (x) + sin (x)) (cos (x) + sin (x)) \cdot 1

乘以:

(cos (x) + sin (x)) \cdot 1 = (cos (x) + sin (x))

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

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

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

分别求解每个部分

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

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

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

- 1 + tan (x) = 0

- 1 + tan (x) = 0

将

1

到右边

- 1 + tan (x) = 0

两边加上

1

- 1 + tan (x) + 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

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

cos (x) + sin (x) = 0

使用三角恒等式改写

cos (x) + sin (x) = 0

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

1 + tan (x) = 0

1 + tan (x) = 0

将

1

到右边

1 + tan (x) = 0

两边减去

1

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

作图

流行的例子

4sin(x)=-cos^2(x)+1

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

sin(2x)=-5cos(2x)

sin (2 x) = - 5 cos (2 x)

tan(x)cot(x)=sec(x)csc(x)

tan (x) cot (x) = sec (x) csc (x)

4sin(x)=cos(x)-2

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

sin(θ)=1-cos(θ)

sin (θ) = 1 - cos (θ)