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受欢迎的三角函数 >

(cos^2(x)+1)/(1+cot^2(x))=1

解答

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

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

1

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} - 1 = 0

化简

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

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

将项转换为分式:

1 = \frac{1 ( 1 + cot ^{2} ( x ) )}{1 + cot ^{2} ( x )}

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

乘以:

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

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

乘开

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

cos^{2} (x) + 1 - (1 + cot^{2} (x))

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

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

打开括号

= - (1) - (cot^{2} (x))

使用加减运算法则

+ (- a) = - a

= - 1 - cot^{2} (x)

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

1 - 1 = 0

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

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

\frac{cos ^{2} ( x ) - cot ^{2} ( x )}{1 + cot ^{2} ( x )} = 0

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

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

分解

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

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

使用平方差公式:

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

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

= (cos (x) + cot (x)) (cos (x) - cot (x))

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

分别求解每个部分

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

cos (x) + cot (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) + cot (x) = 0

用 sin, cos 表示

cos (x) + cot (x)

使用基本三角恒等式:

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

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

化简

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

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

将项转换为分式:

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

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

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

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

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

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

分解

cos (x) + cos (x) sin (x) : cos (x) (sin (x) + 1)

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

因式分解出通项

cos (x)

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

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

分别求解每个部分

cos (x) = 0 or sin (x) + 1 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

的通解

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0

将

1

到右边

sin (x) + 1 = 0

两边减去

1

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

化简

sin (x) = - 1

sin (x) = - 1

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) - cot (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) - cot (x) = 0

用 sin, cos 表示

cos (x) - cot (x)

使用基本三角恒等式:

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

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

化简

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

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

将项转换为分式:

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

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

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

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

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

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

分解

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

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

因式分解出通项

cos (x)

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

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

分别求解每个部分

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

的通解

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

sin (x) - 1 = 0

将

1

到右边

sin (x) - 1 = 0

两边加上

1

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

化简

sin (x) = 1

sin (x) = 1

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

流行的例子

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