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受欢迎的三角函数 >

cos(x)=csc(x)

解答

cos (x) = csc (x)

解答

x \in R 无解

求解步骤

cos (x) = csc (x)

两边减去

csc (x)

cos (x) - csc (x) = 0

用 sin, cos 表示

cos (x) - csc (x)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

= cos (x) - \frac{1}{sin ( x )}

化简

cos (x) - \frac{1}{sin ( x )} : \frac{cos ( x ) sin ( x ) - 1}{sin ( x )}

cos (x) - \frac{1}{sin ( x )}

将项转换为分式:

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

= \frac{cos ( x ) sin ( x )}{sin ( x )} - \frac{1}{sin ( 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{cos ( x ) sin ( x ) - 1}{sin ( x )}

= \frac{cos ( x ) sin ( x ) - 1}{sin ( x )}

\frac{- 1 + cos ( x ) sin ( x )}{sin ( x )} = 0

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

- 1 + cos (x) sin (x) = 0

使用三角恒等式改写

- 1 + cos (x) sin (x)

使用倍角公式:

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

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 1 + \frac{sin ( 2 x )}{2}

- 1 + \frac{sin ( 2 x )}{2} = 0

将

1

到右边

- 1 + \frac{sin ( 2 x )}{2} = 0

两边加上

1

- 1 + \frac{sin ( 2 x )}{2} + 1 = 0 + 1

化简

\frac{sin ( 2 x )}{2} = 1

\frac{sin ( 2 x )}{2} = 1

在两边乘以

2

\frac{sin ( 2 x )}{2} = 1

在两边乘以

2

\frac{2 sin ( 2 x )}{2} = 1 \cdot 2

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R 无解

作图

流行的例子

1+cos(2t)=cos^2(t)

1 + cos (2 t) = cos^{2} (t)

2sin(2x+(3pi)/2)+1=0

2 sin (2 x + \frac{3 π}{2}) + 1 = 0

2cos(2x)-sin(x)-1=0

2 cos (2 x) - sin (x) - 1 = 0

2cos^2(θ)-2sin^2(θ)=sqrt(2)

2 cos^{2} (θ) - 2 sin^{2} (θ) = 2

cot(θ)+3csc(θ)=5

cot (θ) + 3 csc (θ) = 5