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受欢迎的三角函数 >

solvefor w,s(t)=Ae^{-ct}cos(wt+θ)

解答

求解

w, s (t) = A e^{- c t} cos (wt + θ)

解答

w = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}, w = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}

求解步骤

s (t) = A e^{- c t} cos (wt + θ)

交换两边

A e^{- c t} cos (wt + θ) = s t

两边除以

A e^{- c t}

A e^{- c t} cos (wt + θ) = s t

两边除以

A e^{- c t}

\frac{A e ^{- c t} cos ( wt + θ )}{A e ^{- c t}} = \frac{s t}{A e ^{- c t}}

化简

cos (wt + θ) = \frac{s t}{A e ^{- c t}}

cos (wt + θ) = \frac{s t}{A e ^{- c t}}

使用反三角函数性质

cos (wt + θ) = \frac{s t}{A e ^{- c t}}

cos (wt + θ) = \frac{s t}{A e ^{- c t}}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

wt + θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn, wt + θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

wt + θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn, wt + θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

解

wt + θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn : w = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

wt + θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

将

θ

到右边

wt + θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

两边减去

θ

wt + θ - θ = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

化简

wt = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

wt = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

两边除以

t; t \neq = 0

wt = arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

两边除以

t; t \neq = 0

\frac{wt}{t} = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

化简

w = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

w = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

解

wt + θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn : w = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

wt + θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

将

θ

到右边

wt + θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn

两边减去

θ

wt + θ - θ = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

化简

wt = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

wt = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

两边除以

t; t \neq = 0

wt = - arccos (\frac{s t}{A e ^{- c t}}) + 2 πn - θ

两边除以

t; t \neq = 0

\frac{wt}{t} = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

化简

w = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

w = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}; t \neq = 0

w = \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}, w = - \frac{arccos ( \frac{s t}{A e ^{- c t}} )}{t} + \frac{2 πn}{t} - \frac{θ}{t}

作图

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