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 >

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

解答

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

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

2 sec (x)

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

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

合并分式

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

使用法则

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

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

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

sin (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x)

最小公倍数 (LCM)

计算出由出现在

sin (x)

或

cos (x)

中的因子组成的表达式

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

对于

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

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

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

使用三角恒等式改写

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

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

化简

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

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

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

乘开

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

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

使用分配律:

a (b + c) = ab + a c

a = sin (x), b = - 2, c = sin (x)

= sin (x) (- 2) + sin (x) sin (x)

使用加减运算法则

+ (- a) = - a

= - 2 sin (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)

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

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

化简

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

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

对同类项分组

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

同类项相加:

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

= - 2 sin (x) + 1

= - 2 sin (x) + 1

= - 2 sin (x) + 1

1 - 2 sin (x) = 0

将

1

到右边

1 - 2 sin (x) = 0

两边减去

1

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

化简

- 2 sin (x) = - 1

- 2 sin (x) = - 1

两边除以

- 2

- 2 sin (x) = - 1

两边除以

- 2

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

化简

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

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

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

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 = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

2cos(x)+sqrt(3)=0,0<= x<= 2pi

(cot^2(x))/(1+sin(x))=(csc(x)-1)/(sin^2(x))

2sec^2(x)=5tan(x)+5