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受欢迎的三角函数 >

(sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0

解答

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} = 0

解答

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n, x = 75.96375 \dots^{\circ} + 18 0^{\circ} n

求解步骤

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} = 0

化简

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} : \frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )}

将项转换为分式:

5 = \frac{5}{1}

= \frac{sin ( x )}{cos ( x )} - \frac{5}{1} + \frac{4 cos ( x )}{sin ( x )}

cos (x), 1, sin (x)

的最小公倍数:

cos (x) sin (x)

cos (x), 1, sin (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 (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 (x) sin (x)

对于

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

将分母和分子乘以

sin (x)

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

对于

\frac{5}{1} :

将分母和分子乘以

cos (x) sin (x)

\frac{5}{1} = \frac{5 cos ( x ) sin ( x )}{1 \cdot cos ( x ) sin ( x )} = \frac{5 cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

对于

\frac{4 cos ( x )}{sin ( x )} :

将分母和分子乘以

cos (x)

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

= \frac{sin ^{2} ( x )}{cos ( x ) sin ( x )} - \frac{5 cos ( x ) sin ( x )}{cos ( x ) sin ( x )} + \frac{4 cos ^{2} ( 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{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) = 0

分解

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) : (sin (x) - cos (x)) (sin (x) - 4 cos (x))

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x)

将表达式拆分成组

sin^{2} (x) - 5 sin (x) cos (x) + 4 cos^{2} (x)

定义

4

的因数:

1, 2, 4

4

约数 (因数)

找到

4

的质因数:

2, 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2

添加质因数:

2

将 1 和数字

4

自身相加

1, 4

4

的因数

1, 2, 4

4

的负因数:

- 1, - 2, - 4

将因数乘以

- 1

得到负因数

- 1, - 2, - 4

对于每两个因数

u * v = 4

，检验是否

u + v = - 5

检验

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

假检验

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

假

u = - 1, v = - 4

分组为

(a x^{2} + ux y) + (vx y + c y^{2})

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

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

从

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

分解出因式

sin (x)

:

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

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

从

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

分解出因式

- 4 cos (x)

:

- 4 cos (x) (sin (x) - cos (x))

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

使用指数法则:

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

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

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

因式分解出通项

- 4 cos (x)

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

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

因式分解出通项

sin (x) - cos (x)

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

(sin (x) - cos (x)) (sin (x) - 4 cos (x)) = 0

分别求解每个部分

sin (x) - cos (x) = 0 or sin (x) - 4 cos (x) = 0

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

sin (x) - 4 cos (x) = 0 : x = arctan (4) + πn

sin (x) - 4 cos (x) = 0

使用三角恒等式改写

sin (x) - 4 cos (x) = 0

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

tan (x) - 4 = 0

tan (x) - 4 = 0

将

4

到右边

tan (x) - 4 = 0

两边加上

4

tan (x) - 4 + 4 = 0 + 4

化简

tan (x) = 4

tan (x) = 4

使用反三角函数性质

tan (x) = 4

tan (x) = 4

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (4) + πn

x = arctan (4) + πn

合并所有解

x = \frac{π}{4} + πn, x = arctan (4) + πn

以小数形式表示解

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

作图

流行的例子

solvefor x,y=sin(3.2x)

so l v e f or x, y = sin (3.2 x)

sin(x)=0.37,pi<x< pi/2

sin (x) = 0.37, π < x < \frac{π}{2}

5cot(x)+3tan(x)=8

5 cot (x) + 3 tan (x) = 8

sin(2x)-1=cos(2x),\forall 0<= θ<2pi

sin (2 x) - 1 = cos (2 x), \forall 0 \leq θ < 2 π

tan (θ) = \frac{14}{20}