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受欢迎的 三角函数 >

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

  • 初等代数
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  • 微积分
  • 函数
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  • 统计
  • 化学

解答

cos(x)sin(x)​−5+sin(x)4cos(x)​=0

解答

x=4π​+πn,x=1.32581…+πn
+1
度数
x=45∘+180∘n,x=75.96375…∘+180∘n
求解步骤
cos(x)sin(x)​−5+sin(x)4cos(x)​=0
化简 cos(x)sin(x)​−5+sin(x)4cos(x)​:cos(x)sin(x)sin2(x)−5cos(x)sin(x)+4cos2(x)​
cos(x)sin(x)​−5+sin(x)4cos(x)​
将项转换为分式: 5=15​=cos(x)sin(x)​−15​+sin(x)4cos(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 cos(x)sin(x)​:将分母和分子乘以 sin(x)cos(x)sin(x)​=cos(x)sin(x)sin(x)sin(x)​=cos(x)sin(x)sin2(x)​
对于 15​:将分母和分子乘以 cos(x)sin(x)15​=1⋅cos(x)sin(x)5cos(x)sin(x)​=cos(x)sin(x)5cos(x)sin(x)​
对于 sin(x)4cos(x)​:将分母和分子乘以 cos(x)sin(x)4cos(x)​=sin(x)cos(x)4cos(x)cos(x)​=cos(x)sin(x)4cos2(x)​
=cos(x)sin(x)sin2(x)​−cos(x)sin(x)5cos(x)sin(x)​+cos(x)sin(x)4cos2(x)​
因为分母相等,所以合并分式: ca​±cb​=ca±b​=cos(x)sin(x)sin2(x)−5cos(x)sin(x)+4cos2(x)​
cos(x)sin(x)sin2(x)−5cos(x)sin(x)+4cos2(x)​=0
g(x)f(x)​=0⇒f(x)=0sin2(x)−5cos(x)sin(x)+4cos2(x)=0
分解 sin2(x)−5cos(x)sin(x)+4cos2(x):(sin(x)−cos(x))(sin(x)−4cos(x))
sin2(x)−5cos(x)sin(x)+4cos2(x)
将表达式拆分成组
sin2(x)−5sin(x)cos(x)+4cos2(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⋅2=2⋅2
2 是质数,因此无法进一步因数分解=2⋅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⇒假检验 u=2,v=2:u∗v=4,u+v=4⇒假
u=−1,v=−4
分组为 (ax2+uxy)+(vxy+cy2)(sin2(x)−sin(x)cos(x))+(−4sin(x)cos(x)+4cos2(x))
=(sin2(x)−sin(x)cos(x))+(−4sin(x)cos(x)+4cos2(x))
从 sin2(x)−sin(x)cos(x) 分解出因式 sin(x):sin(x)(sin(x)−cos(x))
sin2(x)−sin(x)cos(x)
使用指数法则: ab+c=abacsin2(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 −4sin(x)cos(x)+4cos2(x) 分解出因式 −4cos(x):−4cos(x)(sin(x)−cos(x))
−4sin(x)cos(x)+4cos2(x)
使用指数法则: ab+c=abaccos2(x)=cos(x)cos(x)=−4sin(x)cos(x)+4cos(x)cos(x)
因式分解出通项 −4cos(x)=−4cos(x)(sin(x)−cos(x))
=sin(x)(sin(x)−cos(x))−4cos(x)(sin(x)−cos(x))
因式分解出通项 sin(x)−cos(x)=(sin(x)−cos(x))(sin(x)−4cos(x))
(sin(x)−cos(x))(sin(x)−4cos(x))=0
分别求解每个部分sin(x)−cos(x)=0orsin(x)−4cos(x)=0
sin(x)−cos(x)=0:x=4π​+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)=0cos(x)sin(x)−cos(x)​=cos(x)0​
化简cos(x)sin(x)​−1=0
使用基本三角恒等式: cos(x)sin(x)​=tan(x)tan(x)−1=0
tan(x)−1=0
将 1到右边
tan(x)−1=0
两边加上 1tan(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):
x06π​4π​3π​2π​32π​43π​65π​​tan(x)033​​13​±∞−3​−1−33​​​​
x=4π​+πn
x=4π​+πn
sin(x)−4cos(x)=0:x=arctan(4)+πn
sin(x)−4cos(x)=0
使用三角恒等式改写
sin(x)−4cos(x)=0
在两边除以 cos(x),cos(x)=0cos(x)sin(x)−4cos(x)​=cos(x)0​
化简cos(x)sin(x)​−4=0
使用基本三角恒等式: cos(x)sin(x)​=tan(x)tan(x)−4=0
tan(x)−4=0
将 4到右边
tan(x)−4=0
两边加上 4tan(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⇒x=arctan(a)+πnx=arctan(4)+πn
x=arctan(4)+πn
合并所有解x=4π​+πn,x=arctan(4)+πn
以小数形式表示解x=4π​+πn,x=1.32581…+πn

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solvefor x,y=sin(3.2x)solveforx,y=sin(3.2x)sin(x)=0.37,pi<x< pi/2sin(x)=0.37,π<x<2π​5cot(x)+3tan(x)=85cot(x)+3tan(x)=8sin(2x)-1=cos(2x),\forall 0<= θ<2pisin(2x)−1=cos(2x),∀0≤θ<2πtan(θ)= 14/20tan(θ)=2014​
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