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受欢迎的三角函数 >

tan(5x)+tan(3x)=0

解答

tan (5 x) + tan (3 x) = 0

解答

x = \frac{πn}{4}, x = \frac{π}{8} + \frac{πn}{4}

+1

度数

x = 0^{\circ} + 4 5^{\circ} n, x = 22. 5^{\circ} + 4 5^{\circ} n

求解步骤

tan (5 x) + tan (3 x) = 0

用 sin, cos 表示

tan (3 x) + tan (5 x)

使用基本三角恒等式:

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

= \frac{sin ( 3 x )}{cos ( 3 x )} + tan (5 x)

使用基本三角恒等式:

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

= \frac{sin ( 3 x )}{cos ( 3 x )} + \frac{sin ( 5 x )}{cos ( 5 x )}

化简

\frac{sin ( 3 x )}{cos ( 3 x )} + \frac{sin ( 5 x )}{cos ( 5 x )} : \frac{sin ( 3 x ) cos ( 5 x ) + sin ( 5 x ) cos ( 3 x )}{cos ( 3 x ) cos ( 5 x )}

\frac{sin ( 3 x )}{cos ( 3 x )} + \frac{sin ( 5 x )}{cos ( 5 x )}

cos (3 x), cos (5 x)

的最小公倍数:

cos (3 x) cos (5 x)

cos (3 x), cos (5 x)

最小公倍数 (LCM)

计算出由出现在

cos (3 x)

或

cos (5 x)

中的因子组成的表达式

= cos (3 x) cos (5 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 (3 x) cos (5 x)

对于

\frac{sin ( 3 x )}{cos ( 3 x )} :

将分母和分子乘以

cos (5 x)

\frac{sin ( 3 x )}{cos ( 3 x )} = \frac{sin ( 3 x ) cos ( 5 x )}{cos ( 3 x ) cos ( 5 x )}

对于

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

将分母和分子乘以

cos (3 x)

\frac{sin ( 5 x )}{cos ( 5 x )} = \frac{sin ( 5 x ) cos ( 3 x )}{cos ( 5 x ) cos ( 3 x )}

= \frac{sin ( 3 x ) cos ( 5 x )}{cos ( 3 x ) cos ( 5 x )} + \frac{sin ( 5 x ) cos ( 3 x )}{cos ( 5 x ) cos ( 3 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 ( 3 x ) cos ( 5 x ) + sin ( 5 x ) cos ( 3 x )}{cos ( 3 x ) cos ( 5 x )}

= \frac{sin ( 3 x ) cos ( 5 x ) + sin ( 5 x ) cos ( 3 x )}{cos ( 3 x ) cos ( 5 x )}

\frac{cos ( 3 x ) sin ( 5 x ) + cos ( 5 x ) sin ( 3 x )}{cos ( 3 x ) cos ( 5 x )} = 0

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

cos (3 x) sin (5 x) + cos (5 x) sin (3 x) = 0

使用三角恒等式改写

cos (3 x) sin (5 x) + cos (5 x) sin (3 x)

使用角和恒等式:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= sin (3 x + 5 x)

sin (3 x + 5 x) = 0

sin (3 x + 5 x) = 0

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}

3 x + 5 x = 0 + 2 πn, 3 x + 5 x = π + 2 πn

3 x + 5 x = 0 + 2 πn, 3 x + 5 x = π + 2 πn

解

3 x + 5 x = 0 + 2 πn : x = \frac{πn}{4}

3 x + 5 x = 0 + 2 πn

同类项相加:

3 x + 5 x = 8 x

8 x = 0 + 2 πn

0 + 2 πn = 2 πn

8 x = 2 πn

两边除以

8

8 x = 2 πn

两边除以

8

\frac{8 x}{8} = \frac{2 πn}{8}

化简

x = \frac{πn}{4}

x = \frac{πn}{4}

解

3 x + 5 x = π + 2 πn : x = \frac{π}{8} + \frac{πn}{4}

3 x + 5 x = π + 2 πn

同类项相加:

3 x + 5 x = 8 x

8 x = π + 2 πn

两边除以

8

8 x = π + 2 πn

两边除以

8

\frac{8 x}{8} = \frac{π}{8} + \frac{2 πn}{8}

化简

x = \frac{π}{8} + \frac{πn}{4}

x = \frac{π}{8} + \frac{πn}{4}

x = \frac{πn}{4}, x = \frac{π}{8} + \frac{πn}{4}

作图

流行的例子

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

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

2cos^2(x)-3cos(x)-5=0

2 cos^{2} (x) - 3 cos (x) - 5 = 0

cos(2x)=-cos(x),0<= x<= 2pi

cos (2 x) = - cos (x), 0 \leq x \leq 2 π

2cos^2(w)+3cos(w)+1=0

2 cos^{2} (w) + 3 cos (w) + 1 = 0

cos(3x)+sin(3x)=1

cos (3 x) + sin (3 x) = 1