解答
tan(58x+1∘)⋅cot(2x+7∘)=1
解答
x=113600∘n+3∘,x=166.63636…∘+113600∘n
+1
弧度
x=60π+1120πn,x=660611π+1120πn求解步骤
tan(58x+1∘)cot(2x+7∘)=1
两边减去 1tan(9001440x+180∘)cot(360180x+1260∘)−1=0
用 sin, cos 表示
−1+cot(360180x+1260∘)tan(900180∘+1440x)
使用基本三角恒等式: cot(x)=sin(x)cos(x)=−1+sin(360180x+1260∘)cos(360180x+1260∘)tan(900180∘+1440x)
使用基本三角恒等式: tan(x)=cos(x)sin(x)=−1+sin(360180x+1260∘)cos(360180x+1260∘)⋅cos(900180∘+1440x)sin(900180∘+1440x)
化简 −1+sin(360180x+1260∘)cos(360180x+1260∘)⋅cos(900180∘+1440x)sin(900180∘+1440x):sin(360180x+1260∘)cos(900180∘+1440x)−sin(360180x+1260∘)cos(900180∘+1440x)+cos(360180x+1260∘)sin(900180∘+1440x)
−1+sin(360180x+1260∘)cos(360180x+1260∘)⋅cos(900180∘+1440x)sin(900180∘+1440x)
乘 sin(360180x+1260∘)cos(360180x+1260∘)⋅cos(900180∘+1440x)sin(900180∘+1440x):sin(360180x+1260∘)cos(9001440x+180∘)cos(360180x+1260∘)sin(9001440x+180∘)
sin(360180x+1260∘)cos(360180x+1260∘)⋅cos(900180∘+1440x)sin(900180∘+1440x)
分式相乘: ba⋅dc=b⋅da⋅c=sin(360180x+1260∘)cos(900180∘+1440x)cos(360180x+1260∘)sin(900180∘+1440x)
=−1+sin(360180x+1260∘)cos(9001440x+180∘)cos(360180x+1260∘)sin(9001440x+180∘)
将项转换为分式: 1=sin(360180x+1260∘)cos(900180∘+1440x)1sin(360180x+1260∘)cos(900180∘+1440x)=−sin(360180x+1260∘)cos(900180∘+1440x)1⋅sin(360180x+1260∘)cos(900180∘+1440x)+sin(360180x+1260∘)cos(900180∘+1440x)cos(360180x+1260∘)sin(900180∘+1440x)
因为分母相等，所以合并分式: ca±cb=ca±b=sin(360180x+1260∘)cos(900180∘+1440x)−1⋅sin(360180x+1260∘)cos(900180∘+1440x)+cos(360180x+1260∘)sin(900180∘+1440x)
乘以:1⋅sin(360180x+1260∘)=sin(360180x+1260∘)=sin(360180x+1260∘)cos(9001440x+180∘)−sin(360180x+1260∘)cos(9001440x+180∘)+cos(360180x+1260∘)sin(9001440x+180∘)
=sin(360180x+1260∘)cos(900180∘+1440x)−sin(360180x+1260∘)cos(900180∘+1440x)+cos(360180x+1260∘)sin(900180∘+1440x)
cos(900180∘+1440x)sin(360180x+1260∘)cos(360180x+1260∘)sin(900180∘+1440x)−cos(900180∘+1440x)sin(360180x+1260∘)=0
g(x)f(x)=0⇒f(x)=0cos(360180x+1260∘)sin(900180∘+1440x)−cos(900180∘+1440x)sin(360180x+1260∘)=0
使用三角恒等式改写
cos(360180x+1260∘)sin(900180∘+1440x)−cos(900180∘+1440x)sin(360180x+1260∘)
使用角差恒等式: sin(s)cos(t)−cos(s)sin(t)=sin(s−t)=sin(900180∘+1440x−360180x+1260∘)
sin(900180∘+1440x−360180x+1260∘)=0
sin(900180∘+1440x−360180x+1260∘)=0的通解
sin(x) 周期表（周期为 360∘n"）：
x030∘45∘60∘90∘120∘135∘150∘sin(x)02122231232221x180∘210∘225∘240∘270∘300∘315∘330∘sin(x)0−21−22−23−1−23−22−21
900180∘+1440x−360180x+1260∘=0+360∘n,900180∘+1440x−360180x+1260∘=180∘+360∘n
900180∘+1440x−360180x+1260∘=0+360∘n,900180∘+1440x−360180x+1260∘=180∘+360∘n
解 900180∘+1440x−360180x+1260∘=0+360∘n:x=113600∘n+3∘
900180∘+1440x−360180x+1260∘=0+360∘n
0+360∘n=360∘n900180∘+1440x−360180x+1260∘=360∘n
乘以最小公倍数
900180∘+1440x−360180x+1260∘=360∘n
找到 900,360 的最小公倍数:1800
900,360
最小公倍数 (LCM)
900质因数分解:2⋅2⋅3⋅3⋅5⋅5
900
900除以 2900=450⋅2=2⋅450
450除以 2450=225⋅2=2⋅2⋅225
225除以 3225=75⋅3=2⋅2⋅3⋅75
75除以 375=25⋅3=2⋅2⋅3⋅3⋅25
25除以 525=5⋅5=2⋅2⋅3⋅3⋅5⋅5
2,3,5 都是质数，因此无法进一步因数分解=2⋅2⋅3⋅3⋅5⋅5
360质因数分解:2⋅2⋅2⋅3⋅3⋅5
360
360除以 2360=180⋅2=2⋅180
180除以 2180=90⋅2=2⋅2⋅90
90除以 290=45⋅2=2⋅2⋅2⋅45
45除以 345=15⋅3=2⋅2⋅2⋅3⋅15
15除以 315=5⋅3=2⋅2⋅2⋅3⋅3⋅5
2,3,5 都是质数，因此无法进一步因数分解=2⋅2⋅2⋅3⋅3⋅5
将每个因子乘以它在 900 或 360中出现的最多次数=2⋅2⋅2⋅3⋅3⋅5⋅5
数字相乘:2⋅2⋅2⋅3⋅3⋅5⋅5=1800=1800
乘以最小公倍数=1800900180∘+1440x⋅1800−360180x+1260∘⋅1800=360∘n⋅1800
化简
900180∘+1440x⋅1800−360180x+1260∘⋅1800=360∘n⋅1800
化简 900180∘+1440x⋅1800:2(1440x+180∘)
900180∘+1440x⋅1800
分式相乘: a⋅cb=ca⋅b=900(180∘+1440x)⋅1800
数字相除:9001800=2=2(1440x+180∘)
化简 −360180x+1260∘⋅1800:−5(180x+1260∘)
−360180x+1260∘⋅1800
分式相乘: a⋅cb=ca⋅b=−360(180x+1260∘)⋅1800
数字相除:3601800=5=−5(180x+1260∘)
化简 360∘n⋅1800:648000∘n
360∘n⋅1800
数字相乘:2⋅1800=3600=648000∘n
2(1440x+180∘)−5(180x+1260∘)=648000∘n
2(1440x+180∘)−5(180x+1260∘)=648000∘n
2(1440x+180∘)−5(180x+1260∘)=648000∘n
展开 2(1440x+180∘)−5(180x+1260∘):1980x−5940∘
2(1440x+180∘)−5(180x+1260∘)
乘开 2(1440x+180∘):2880x+360∘
2(1440x+180∘)
使用分配律: a(b+c)=ab+aca=2,b=1440x,c=180∘=2⋅1440x+360∘
数字相乘:2⋅1440=2880=2880x+360∘
=2880x+360∘−5(180x+1260∘)
乘开 −5(180x+1260∘):−900x−6300∘
−5(180x+1260∘)
使用分配律: a(b+c)=ab+aca=−5,b=180x,c=1260∘=−5⋅180x+(−5)⋅1260∘
使用加减运算法则+(−a)=−a=−5⋅180x−5⋅1260∘
化简 −5⋅180x−5⋅1260∘:−900x−6300∘
−5⋅180x−5⋅1260∘
数字相乘:5⋅180=900=−900x−5⋅1260∘
数字相乘:5⋅7=35=−900x−6300∘
=−900x−6300∘
=2880x+360∘−900x−6300∘
化简 2880x+360∘−900x−6300∘:1980x−5940∘
2880x+360∘−900x−6300∘
对同类项分组=2880x−900x+360∘−6300∘
同类项相加:2880x−900x=1980x=1980x+360∘−6300∘
同类项相加:360∘−6300∘=−5940∘=1980x−5940∘
=1980x−5940∘
1980x−5940∘=648000∘n
将 5940∘到右边
1980x−5940∘=648000∘n
两边加上 5940∘1980x−5940∘+5940∘=648000∘n+5940∘
化简1980x=648000∘n+5940∘
1980x=648000∘n+5940∘
两边除以 1980
1980x=648000∘n+5940∘
两边除以 198019801980x=1980648000∘n+3∘
化简
19801980x=1980648000∘n+3∘
化简 19801980x:x
19801980x
数字相除:19801980=1=x
化简 1980648000∘n+3∘:113600∘n+3∘
1980648000∘n+3∘
消掉 1980648000∘n:113600∘n
1980648000∘n
约分:180=113600∘n
=113600∘n+3∘
消掉 3∘:3∘
3∘
约分:33=3∘
=113600∘n+3∘
x=113600∘n+3∘
x=113600∘n+3∘
x=113600∘n+3∘
解 900180∘+1440x−360180x+1260∘=180∘+360∘n:x=166.63636…∘+113600∘n
900180∘+1440x−360180x+1260∘=180∘+360∘n
乘以最小公倍数
900180∘+1440x−360180x+1260∘=180∘+360∘n
找到 900,360 的最小公倍数:1800
900,360
最小公倍数 (LCM)
900质因数分解:2⋅2⋅3⋅3⋅5⋅5
900
900除以 2900=450⋅2=2⋅450
450除以 2450=225⋅2=2⋅2⋅225
225除以 3225=75⋅3=2⋅2⋅3⋅75
75除以 375=25⋅3=2⋅2⋅3⋅3⋅25
25除以 525=5⋅5=2⋅2⋅3⋅3⋅5⋅5
2,3,5 都是质数，因此无法进一步因数分解=2⋅2⋅3⋅3⋅5⋅5
360质因数分解:2⋅2⋅2⋅3⋅3⋅5
360
360除以 2360=180⋅2=2⋅180
180除以 2180=90⋅2=2⋅2⋅90
90除以 290=45⋅2=2⋅2⋅2⋅45
45除以 345=15⋅3=2⋅2⋅2⋅3⋅15
15除以 315=5⋅3=2⋅2⋅2⋅3⋅3⋅5
2,3,5 都是质数，因此无法进一步因数分解=2⋅2⋅2⋅3⋅3⋅5
将每个因子乘以它在 900 或 360中出现的最多次数=2⋅2⋅2⋅3⋅3⋅5⋅5
数字相乘:2⋅2⋅2⋅3⋅3⋅5⋅5=1800=1800
乘以最小公倍数=1800900180∘+1440x⋅1800−360180x+1260∘⋅1800=180∘1800+360∘n⋅1800
化简
900180∘+1440x⋅1800−360180x+1260∘⋅1800=180∘1800+360∘n⋅1800
化简 900180∘+1440x⋅1800:2(1440x+180∘)
900180∘+1440x⋅1800
分式相乘: a⋅cb=ca⋅b=900(180∘+1440x)⋅1800
数字相除:9001800=2=2(1440x+180∘)
化简 −360180x+1260∘⋅1800:−5(180x+1260∘)
−360180x+1260∘⋅1800
分式相乘: a⋅cb=ca⋅b=−360(180x+1260∘)⋅1800
数字相除:3601800=5=−5(180x+1260∘)
化简 180∘1800:324000∘
180∘1800
使用交换律:180∘1800=324000∘324000∘
化简 360∘n⋅1800:648000∘n
360∘n⋅1800
数字相乘:2⋅1800=3600=648000∘n
2(1440x+180∘)−5(180x+1260∘)=324000∘+648000∘n
2(1440x+180∘)−5(180x+1260∘)=324000∘+648000∘n
2(1440x+180∘)−5(180x+1260∘)=324000∘+648000∘n
展开 2(1440x+180∘)−5(180x+1260∘):1980x−5940∘
2(1440x+180∘)−5(180x+1260∘)
乘开 2(1440x+180∘):2880x+360∘
2(1440x+180∘)
使用分配律: a(b+c)=ab+aca=2,b=1440x,c=180∘=2⋅1440x+360∘
数字相乘:2⋅1440=2880=2880x+360∘
=2880x+360∘−5(180x+1260∘)
乘开 −5(180x+1260∘):−900x−6300∘
−5(180x+1260∘)
使用分配律: a(b+c)=ab+aca=−5,b=180x,c=1260∘=−5⋅180x+(−5)⋅1260∘
使用加减运算法则+(−a)=−a=−5⋅180x−5⋅1260∘
化简 −5⋅180x−5⋅1260∘:−900x−6300∘
−5⋅180x−5⋅1260∘
数字相乘:5⋅180=900=−900x−5⋅1260∘
数字相乘:5⋅7=35=−900x−6300∘
=−900x−6300∘
=2880x+360∘−900x−6300∘
化简 2880x+360∘−900x−6300∘:1980x−5940∘
2880x+360∘−900x−6300∘
对同类项分组=2880x−900x+360∘−6300∘
同类项相加:2880x−900x=1980x=1980x+360∘−6300∘
同类项相加:360∘−6300∘=−5940∘=1980x−5940∘
=1980x−5940∘
1980x−5940∘=324000∘+648000∘n
将 5940∘到右边
1980x−5940∘=324000∘+648000∘n
两边加上 5940∘1980x−5940∘+5940∘=324000∘+648000∘n+5940∘
化简1980x=329940∘+648000∘n
1980x=329940∘+648000∘n
两边除以 1980
1980x=329940∘+648000∘n
两边除以 198019801980x=166.63636…∘+1980648000∘n
化简
19801980x=166.63636…∘+1980648000∘n
化简 19801980x:x
19801980x
数字相除:19801980=1=x
化简 166.63636…∘+1980648000∘n:166.63636…∘+113600∘n
166.63636…∘+1980648000∘n
消掉 166.63636…∘:166.63636…∘
166.63636…∘
约分:3=166.63636…∘
=166.63636…∘+1980648000∘n
消掉 1980648000∘n:113600∘n
1980648000∘n
约分:180=113600∘n
=166.63636…∘+113600∘n
x=166.63636…∘+113600∘n
x=166.63636…∘+113600∘n
x=166.63636…∘+113600∘n
x=113600∘n+3∘,x=166.63636…∘+113600∘n