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受欢迎的三角函数 >

5cot(x)=cos(x)

解答

5 cot (x) = cos (x)

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

5 cot (x) = cos (x)

两边减去

cos (x)

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

用 sin, cos 表示

- cos (x) + 5 cot (x)

使用基本三角恒等式:

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

= - cos (x) + 5 \cdot \frac{cos ( x )}{sin ( x )}

化简

- cos (x) + 5 \cdot \frac{cos ( x )}{sin ( x )} : \frac{- cos ( x ) sin ( x ) + 5 cos ( x )}{sin ( x )}

- cos (x) + 5 \cdot \frac{cos ( x )}{sin ( x )}

乘

5 \cdot \frac{cos ( x )}{sin ( x )} : \frac{5 cos ( x )}{sin ( x )}

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

分式相乘:

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

= \frac{cos ( x ) \cdot 5}{sin ( x )}

= - cos (x) + \frac{5 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 ) \cdot 5}{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 ) \cdot 5}{sin ( x )}

= \frac{- cos ( x ) sin ( x ) + 5 cos ( x )}{sin ( x )}

\frac{5 cos ( x ) - cos ( x ) sin ( x )}{sin ( x )} = 0

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

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

分解

5 cos (x) - cos (x) sin (x) : - cos (x) (sin (x) - 5)

5 cos (x) - cos (x) sin (x)

因式分解出通项

- cos (x)

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

- cos (x) (sin (x) - 5) = 0

分别求解每个部分

cos (x) = 0 or sin (x) - 5 = 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) - 5 = 0 :

无解

sin (x) - 5 = 0

将

5

到右边

sin (x) - 5 = 0

两边加上

5

sin (x) - 5 + 5 = 0 + 5

化简

sin (x) = 5

sin (x) = 5

- 1 \leq sin (x) \leq 1

无解

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

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

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

9sin^2(θ)+19sin(θ)+8=7sin(θ)+4

9 sin^{2} (θ) + 19 sin (θ) + 8 = 7 sin (θ) + 4

arccos(x)=(7pi)/8

arccos (x) = \frac{7 π}{8}

(sin(37))/(23)=(sin(x))/(21)

\frac{sin ( 3 7 ^{\circ} )}{23} = \frac{sin ( x )}{21}

csc (x) = \frac{1}{3}