解答
tan(θ+10∘)=cot(2θ−10)
解答
θ=26.66666…∘+310+3360∘n,θ=86.66666…∘+310+3360∘n
+1
弧度
θ=274π+310+32πn,θ=2713π+310+32πn求解步骤
tan(θ+10∘)=cot(2θ−10)
两边减去 cot(2θ−10)tan(θ+10∘)−cot(2θ−10)=0
用 sin, cos 表示
−cot(−10+2θ)+tan(10∘+θ)
使用基本三角恒等式: cot(x)=sin(x)cos(x)=−sin(−10+2θ)cos(−10+2θ)+tan(10∘+θ)
使用基本三角恒等式: tan(x)=cos(x)sin(x)=−sin(−10+2θ)cos(−10+2θ)+cos(10∘+θ)sin(10∘+θ)
化简 −sin(−10+2θ)cos(−10+2θ)+cos(10∘+θ)sin(10∘+θ):sin(2θ−10)cos(1818θ+180∘)−cos(−10+2θ)cos(1818θ+180∘)+sin(18180∘+18θ)sin(2θ−10)
−sin(−10+2θ)cos(−10+2θ)+cos(10∘+θ)sin(10∘+θ)
cos(10∘+θ)sin(10∘+θ)=cos(18180∘+18θ)sin(18180∘+18θ)
cos(10∘+θ)sin(10∘+θ)
化简 10∘+θ:18180∘+18θ
10∘+θ
将项转换为分式: θ=18θ18=10∘+18θ⋅18
因为分母相等，所以合并分式: ca±cb=ca±b=18180∘+θ⋅18
=cos(18180∘+θ⋅18)sin(10∘+θ)
化简 10∘+θ:18180∘+18θ
10∘+θ
将项转换为分式: θ=18θ18=10∘+18θ⋅18
因为分母相等，所以合并分式: ca±cb=ca±b=18180∘+θ⋅18
=cos(18180∘+θ⋅18)sin(18180∘+θ⋅18)
=−sin(2θ−10)cos(2θ−10)+cos(1818θ+180∘)sin(1818θ+180∘)
sin(−10+2θ),cos(18180∘+θ18)的最小公倍数:sin(2θ−10)cos(1818θ+180∘)
sin(−10+2θ),cos(18180∘+θ⋅18)
最小公倍数 (LCM)
计算出由出现在 sin(−10+2θ) 或 cos(18180∘+θ18)中的因子组成的表达式=sin(2θ−10)cos(1818θ+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(2θ−10)cos(1818θ+180∘)
对于 sin(−10+2θ)cos(−10+2θ):将分母和分子乘以 cos(1818θ+180∘)sin(−10+2θ)cos(−10+2θ)=sin(−10+2θ)cos(1818θ+180∘)cos(−10+2θ)cos(1818θ+180∘)
对于 cos(18180∘+θ⋅18)sin(18180∘+θ⋅18):将分母和分子乘以 sin(2θ−10)cos(18180∘+θ⋅18)sin(18180∘+θ⋅18)=cos(18180∘+θ⋅18)sin(2θ−10)sin(18180∘+θ⋅18)sin(2θ−10)
=−sin(−10+2θ)cos(1818θ+180∘)cos(−10+2θ)cos(1818θ+180∘)+cos(18180∘+θ⋅18)sin(2θ−10)sin(18180∘+θ⋅18)sin(2θ−10)
因为分母相等，所以合并分式: ca±cb=ca±b=sin(2θ−10)cos(1818θ+180∘)−cos(−10+2θ)cos(1818θ+180∘)+sin(18180∘+θ⋅18)sin(2θ−10)
=sin(2θ−10)cos(1818θ+180∘)−cos(−10+2θ)cos(1818θ+180∘)+sin(18180∘+18θ)sin(2θ−10)
cos(18180∘+18θ)sin(−10+2θ)−cos(−10+2θ)cos(18180∘+18θ)+sin(−10+2θ)sin(18180∘+18θ)=0
g(x)f(x)=0⇒f(x)=0−cos(−10+2θ)cos(18180∘+18θ)+sin(−10+2θ)sin(18180∘+18θ)=0
使用三角恒等式改写
−cos(−10+2θ)cos(18180∘+18θ)+sin(−10+2θ)sin(18180∘+18θ)
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(−10+2θ+18180∘+18θ)
−cos(−10+2θ+18180∘+18θ)=0
两边除以 −1
−cos(−10+2θ+18180∘+18θ)=0
两边除以 −1−1−cos(−10+2θ+18180∘+18θ)=−10
化简cos(−10+2θ+18180∘+18θ)=0
cos(−10+2θ+18180∘+18θ)=0
cos(−10+2θ+18180∘+18θ)=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
−10+2θ+18180∘+18θ=90∘+360∘n,−10+2θ+18180∘+18θ=270∘+360∘n
−10+2θ+18180∘+18θ=90∘+360∘n,−10+2θ+18180∘+18θ=270∘+360∘n
解 −10+2θ+18180∘+18θ=90∘+360∘n:θ=26.66666…∘+310+3360∘n
−10+2θ+18180∘+18θ=90∘+360∘n
将 10到右边
−10+2θ+18180∘+18θ=90∘+360∘n
两边加上 10−10+2θ+18180∘+18θ+10=90∘+360∘n+10
化简2θ+18180∘+18θ=90∘+360∘n+10
2θ+18180∘+18θ=90∘+360∘n+10
在两边乘以 18
2θ+18180∘+18θ=90∘+360∘n+10
在两边乘以 182θ⋅18+18180∘+18θ⋅18=90∘⋅18+360∘n⋅18+10⋅18
化简
2θ⋅18+18180∘+18θ⋅18=90∘⋅18+360∘n⋅18+10⋅18
化简 2θ⋅18:36θ
2θ⋅18
数字相乘:2⋅18=36=36θ
化简 18180∘+18θ⋅18:180∘+18θ
18180∘+18θ⋅18
分式相乘: a⋅cb=ca⋅b=18(180∘+18θ)⋅18
约分:18=180∘+18θ
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
化简 10⋅18:180
10⋅18
数字相乘:10⋅18=180=180
36θ+180∘+18θ=1620∘+6480∘n+180
54θ+180∘=1620∘+6480∘n+180
54θ+180∘=1620∘+6480∘n+180
54θ+180∘=1620∘+6480∘n+180
将 180∘到右边
54θ+180∘=1620∘+6480∘n+180
两边减去 180∘54θ+180∘−180∘=1620∘+6480∘n+180−180∘
化简54θ=1440∘+6480∘n+180
54θ=1440∘+6480∘n+180
两边除以 54
54θ=1440∘+6480∘n+180
两边除以 545454θ=26.66666…∘+546480∘n+54180
化简
5454θ=26.66666…∘+546480∘n+54180
化简 5454θ:θ
5454θ
数字相除:5454=1=θ
化简 26.66666…∘+546480∘n+54180:26.66666…∘+310+3360∘n
26.66666…∘+546480∘n+54180
对同类项分组=26.66666…∘+54180+546480∘n
消掉 26.66666…∘:26.66666…∘
26.66666…∘
约分:2=26.66666…∘
=26.66666…∘+54180+546480∘n
消掉 54180:310
54180
约分:18=310
=26.66666…∘+310+546480∘n
消掉 546480∘n:3360∘n
546480∘n
约分:18=3360∘n
=26.66666…∘+310+3360∘n
θ=26.66666…∘+310+3360∘n
θ=26.66666…∘+310+3360∘n
θ=26.66666…∘+310+3360∘n
解 −10+2θ+18180∘+18θ=270∘+360∘n:θ=86.66666…∘+310+3360∘n
−10+2θ+18180∘+18θ=270∘+360∘n
将 10到右边
−10+2θ+18180∘+18θ=270∘+360∘n
两边加上 10−10+2θ+18180∘+18θ+10=270∘+360∘n+10
化简2θ+18180∘+18θ=270∘+360∘n+10
2θ+18180∘+18θ=270∘+360∘n+10
在两边乘以 18
2θ+18180∘+18θ=270∘+360∘n+10
在两边乘以 182θ⋅18+18180∘+18θ⋅18=270∘⋅18+360∘n⋅18+10⋅18
化简
2θ⋅18+18180∘+18θ⋅18=270∘⋅18+360∘n⋅18+10⋅18
化简 2θ⋅18:36θ
2θ⋅18
数字相乘:2⋅18=36=36θ
化简 18180∘+18θ⋅18:180∘+18θ
18180∘+18θ⋅18
分式相乘: a⋅cb=ca⋅b=18(180∘+18θ)⋅18
约分:18=180∘+18θ
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
化简 10⋅18:180
10⋅18
数字相乘:10⋅18=180=180
36θ+180∘+18θ=4860∘+6480∘n+180
54θ+180∘=4860∘+6480∘n+180
54θ+180∘=4860∘+6480∘n+180
54θ+180∘=4860∘+6480∘n+180
将 180∘到右边
54θ+180∘=4860∘+6480∘n+180
两边减去 180∘54θ+180∘−180∘=4860∘+6480∘n+180−180∘
化简54θ=4680∘+6480∘n+180
54θ=4680∘+6480∘n+180
两边除以 54
54θ=4680∘+6480∘n+180
两边除以 545454θ=86.66666…∘+546480∘n+54180
化简
5454θ=86.66666…∘+546480∘n+54180
化简 5454θ:θ
5454θ
数字相除:5454=1=θ
化简 86.66666…∘+546480∘n+54180:86.66666…∘+310+3360∘n
86.66666…∘+546480∘n+54180
对同类项分组=86.66666…∘+54180+546480∘n
消掉 86.66666…∘:86.66666…∘
86.66666…∘
约分:2=86.66666…∘
=86.66666…∘+54180+546480∘n
消掉 54180:310
54180
约分:18=310
=86.66666…∘+310+546480∘n
消掉 546480∘n:3360∘n
546480∘n
约分:18=3360∘n
=86.66666…∘+310+3360∘n
θ=86.66666…∘+310+3360∘n
θ=86.66666…∘+310+3360∘n
θ=86.66666…∘+310+3360∘n
θ=26.66666…∘+310+3360∘n,θ=86.66666…∘+310+3360∘n