解答
tan(β+10∘)=cot(2β−10∘)
解答
β=30∘+3360∘n,β=90∘+3360∘n
+1
弧度
β=6π+32πn,β=2π+32πn求解步骤
tan(β+10∘)=cot(2β−10∘)
两边减去 cot(2β−10∘)tan(β+10∘)−cot(2β−10∘)=0
化简 tan(β+10∘)−cot(2β−10∘):tan(1818β+180∘)−cot(1836β−180∘)
tan(β+10∘)−cot(2β−10∘)
化简 β+10∘:1818β+180∘
β+10∘
将项转换为分式: β=18β18=18β⋅18+10∘
因为分母相等，所以合并分式: ca±cb=ca±b=18β⋅18+180∘
=tan(1818β+180∘)−cot(2β−10∘)
化简 2β−10∘:1836β−180∘
2β−10∘
将项转换为分式: 2β=182β18=182β⋅18−10∘
因为分母相等，所以合并分式: ca±cb=ca±b=182β⋅18−180∘
数字相乘:2⋅18=36=1836β−180∘
=tan(1818β+180∘)−cot(1836β−180∘)
tan(1818β+180∘)−cot(1836β−180∘)=0
用 sin, cos 表示
−cot(18−180∘+36β)+tan(18180∘+18β)
使用基本三角恒等式: cot(x)=sin(x)cos(x)=−sin(18−180∘+36β)cos(18−180∘+36β)+tan(18180∘+18β)
使用基本三角恒等式: tan(x)=cos(x)sin(x)=−sin(18−180∘+36β)cos(18−180∘+36β)+cos(18180∘+18β)sin(18180∘+18β)
化简 −sin(18−180∘+36β)cos(18−180∘+36β)+cos(18180∘+18β)sin(18180∘+18β):sin(1836β−180∘)cos(1818β+180∘)−cos(18−180∘+36β)cos(1818β+180∘)+sin(18180∘+18β)sin(1836β−180∘)
−sin(18−180∘+36β)cos(18−180∘+36β)+cos(18180∘+18β)sin(18180∘+18β)
sin(18−180∘+36β),cos(18180∘+18β)的最小公倍数:sin(1836β−180∘)cos(1818β+180∘)
sin(18−180∘+36β),cos(18180∘+18β)
最小公倍数 (LCM)
计算出由出现在 sin(18−180∘+36β) 或 cos(18180∘+18β)中的因子组成的表达式=sin(1836β−180∘)cos(1818β+180∘)
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(1836β−180∘)cos(1818β+180∘)
对于 sin(18−180∘+36β)cos(18−180∘+36β):将分母和分子乘以 cos(1818β+180∘)sin(18−180∘+36β)cos(18−180∘+36β)=sin(18−180∘+36β)cos(1818β+180∘)cos(18−180∘+36β)cos(1818β+180∘)
对于 cos(18180∘+18β)sin(18180∘+18β):将分母和分子乘以 sin(1836β−180∘)cos(18180∘+18β)sin(18180∘+18β)=cos(18180∘+18β)sin(1836β−180∘)sin(18180∘+18β)sin(1836β−180∘)
=−sin(18−180∘+36β)cos(1818β+180∘)cos(18−180∘+36β)cos(1818β+180∘)+cos(18180∘+18β)sin(1836β−180∘)sin(18180∘+18β)sin(1836β−180∘)
因为分母相等，所以合并分式: ca±cb=ca±b=sin(1836β−180∘)cos(1818β+180∘)−cos(18−180∘+36β)cos(1818β+180∘)+sin(18180∘+18β)sin(1836β−180∘)
=sin(1836β−180∘)cos(1818β+180∘)−cos(18−180∘+36β)cos(1818β+180∘)+sin(18180∘+18β)sin(1836β−180∘)
cos(18180∘+18β)sin(18−180∘+36β)−cos(18−180∘+36β)cos(18180∘+18β)+sin(18−180∘+36β)sin(18180∘+18β)=0
g(x)f(x)=0⇒f(x)=0−cos(18−180∘+36β)cos(18180∘+18β)+sin(18−180∘+36β)sin(18180∘+18β)=0
使用三角恒等式改写
−cos(18−180∘+36β)cos(18180∘+18β)+sin(18−180∘+36β)sin(18180∘+18β)
使用角和恒等式: cos(s)cos(t)−sin(s)sin(t)=cos(s+t)−cos(s)cos(t)+sin(s)sin(t)=−cos(s+t)=−cos(18−180∘+36β+18180∘+18β)
−cos(18−180∘+36β+18180∘+18β)=0
两边除以 −1
−cos(18−180∘+36β+18180∘+18β)=0
两边除以 −1−1−cos(18−180∘+36β+18180∘+18β)=−10
化简cos(18−180∘+36β+18180∘+18β)=0
cos(18−180∘+36β+18180∘+18β)=0
cos(18−180∘+36β+18180∘+18β)=0的通解
cos(x) 周期表（周期为 360∘n）：
x030∘45∘60∘90∘120∘135∘150∘cos(x)12322210−21−22−23x180∘210∘225∘240∘270∘300∘315∘330∘cos(x)−1−23−22−210212223
18−180∘+36β+18180∘+18β=90∘+360∘n,18−180∘+36β+18180∘+18β=270∘+360∘n
18−180∘+36β+18180∘+18β=90∘+360∘n,18−180∘+36β+18180∘+18β=270∘+360∘n
解 18−180∘+36β+18180∘+18β=90∘+360∘n:β=30∘+3360∘n
18−180∘+36β+18180∘+18β=90∘+360∘n
在两边乘以 18
18−180∘+36β+18180∘+18β=90∘+360∘n
在两边乘以 1818−180∘+36β⋅18+18180∘+18β⋅18=90∘⋅18+360∘n⋅18
化简
18−180∘+36β⋅18+18180∘+18β⋅18=90∘⋅18+360∘n⋅18
化简 18−180∘+36β⋅18:−180∘+36β
18−180∘+36β⋅18
分式相乘: a⋅cb=ca⋅b=18(−180∘+36β)⋅18
约分:18=−−180∘+36β
化简 18180∘+18β⋅18:180∘+18β
18180∘+18β⋅18
分式相乘: a⋅cb=ca⋅b=18(180∘+18β)⋅18
约分:18=180∘+18β
化简 90∘⋅18:1620∘
90∘⋅18
分式相乘: a⋅cb=ca⋅b=1620∘
数字相除:218=9=1620∘
化简 360∘n⋅18:6480∘n
360∘n⋅18
数字相乘:2⋅18=36=6480∘n
−180∘+36β+180∘+18β=1620∘+6480∘n
54β=1620∘+6480∘n
54β=1620∘+6480∘n
54β=1620∘+6480∘n
两边除以 54
54β=1620∘+6480∘n
两边除以 545454β=30∘+546480∘n
化简
5454β=30∘+546480∘n
化简 5454β:β
5454β
数字相除:5454=1=β
化简 30∘+546480∘n:30∘+3360∘n
30∘+546480∘n
消掉 30∘:30∘
30∘
约分:9=30∘
=30∘+546480∘n
消掉 546480∘n:3360∘n
546480∘n
约分:18=3360∘n
=30∘+3360∘n
β=30∘+3360∘n
β=30∘+3360∘n
β=30∘+3360∘n
解 18−180∘+36β+18180∘+18β=270∘+360∘n:β=90∘+3360∘n
18−180∘+36β+18180∘+18β=270∘+360∘n
在两边乘以 18
18−180∘+36β+18180∘+18β=270∘+360∘n
在两边乘以 1818−180∘+36β⋅18+18180∘+18β⋅18=270∘⋅18+360∘n⋅18
化简
18−180∘+36β⋅18+18180∘+18β⋅18=270∘⋅18+360∘n⋅18
化简 18−180∘+36β⋅18:−180∘+36β
18−180∘+36β⋅18
分式相乘: a⋅cb=ca⋅b=18(−180∘+36β)⋅18
约分:18=−−180∘+36β
化简 18180∘+18β⋅18:180∘+18β
18180∘+18β⋅18
分式相乘: a⋅cb=ca⋅b=18(180∘+18β)⋅18
约分:18=180∘+18β
化简 270∘⋅18:4860∘
270∘⋅18
分式相乘: a⋅cb=ca⋅b=4860∘
数字相乘:3⋅18=54=4860∘
数字相除:254=27=4860∘
化简 360∘n⋅18:6480∘n
360∘n⋅18
数字相乘:2⋅18=36=6480∘n
−180∘+36β+180∘+18β=4860∘+6480∘n
54β=4860∘+6480∘n
54β=4860∘+6480∘n
54β=4860∘+6480∘n
两边除以 54
54β=4860∘+6480∘n
两边除以 545454β=90∘+546480∘n
化简
5454β=90∘+546480∘n
化简 5454β:β
5454β
数字相除:5454=1=β
化简 90∘+546480∘n:90∘+3360∘n
90∘+546480∘n
消掉 90∘:90∘
90∘
约分:27=90∘
=90∘+546480∘n
消掉 546480∘n:3360∘n
546480∘n
约分:18=3360∘n
=90∘+3360∘n
β=90∘+3360∘n
β=90∘+3360∘n
β=90∘+3360∘n
β=30∘+3360∘n,β=90∘+3360∘n