What is the general solution for solvefor g,θ(t)=-1cos(sqrt(g/l)t) ?

The general solution for solvefor g,θ(t)=-1cos(sqrt(g/l)t) is

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solvefor g,θ(t)=-1cos(sqrt(g/l)t)

Solution

solve for

g, θ (t) = - 1 cos (\frac{g}{l} t)

Solution

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}, g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

Solution steps

θ (t) = - 1 \cdot cos (\frac{g}{l} t)

Switch sides

- 1 \cdot cos (\frac{g}{l} t) = θt

Divide both sides by

- 1

- 1 \cdot cos (\frac{g}{l} t) = θt

Divide both sides by

- 1

\frac{- 1 \cdot cos ( \frac{g}{l} t )}{- 1} = \frac{θt}{- 1}

Simplify

cos (\frac{g}{l} t) = - θt

cos (\frac{g}{l} t) = - θt

Apply trig inverse properties

cos (\frac{g}{l} t) = - θt

General solutions for

cos (\frac{g}{l} t) = - θt

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

\frac{g}{l} t = arccos (- θt) + 2 πn, \frac{g}{l} t = - arccos (- θt) + 2 πn

\frac{g}{l} t = arccos (- θt) + 2 πn, \frac{g}{l} t = - arccos (- θt) + 2 πn

Solve

\frac{g}{l} t = arccos (- θt) + 2 πn : g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{g}{l} t = arccos (- θt) + 2 πn

Divide both sides by

t

\frac{g}{l} t = arccos (- θt) + 2 πn

Divide both sides by

t

\frac{\frac{g}{l} t}{t} = \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

Simplify

\frac{g}{l} = \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

\frac{g}{l} = \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

Square both sides:

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

(\frac{g}{l})^{2} = (\frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2}

Expand

(\frac{g}{l})^{2} : \frac{g}{l}

(\frac{g}{l})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((\frac{g}{l})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (\frac{g}{l})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= \frac{g}{l}

Expand

(\frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2} : \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

(\frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2}

Combine the fractions

\frac{arccos ( - θt )}{t} + \frac{2 πn}{t} : \frac{arccos ( - θt ) + 2 πn}{t}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{arccos ( - θt ) + 2 πn}{t}

= (\frac{arccos ( - θt ) + 2 πn}{t})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( arccos ( - θt ) + 2 πn ) ^{2}}{t ^{2}}

(arccos (- θt) + 2 πn)^{2} = arccos^{2} (- θt) + 4 πn arccos (- θt) + 4 π^{2} n^{2}

(arccos (- θt) + 2 πn)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = arccos (- θt), b = 2 πn

= arccos^{2} (- θt) + 2 arccos (- θt) \cdot 2 πn + (2 πn)^{2}

Simplify

arccos^{2} (- θt) + 2 arccos (- θt) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (- θt) + 4 πn arccos (- θt) + 4 π^{2} n^{2}

arccos^{2} (- θt) + 2 arccos (- θt) \cdot 2 πn + (2 πn)^{2}

2 arccos (- θt) \cdot 2 πn = 4 πn arccos (- θt)

2 arccos (- θt) \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn arccos (- θt)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (- θt) + 4 πn arccos (- θt) + 4 π^{2} n^{2}

= arccos^{2} (- θt) + 4 πn arccos (- θt) + 4 π^{2} n^{2}

= \frac{arccos ^{2} ( - θt ) + 4 πn arccos ( - θt ) + 4 π ^{2} n ^{2}}{t ^{2}}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{arccos ^{2} ( - θt ) + 4 πn arccos ( - θt ) + 4 π ^{2} n ^{2}}{t ^{2}} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

= \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Solve

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}} : g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply both sides by

l

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply both sides by

l

\frac{g l}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} l + \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Simplify

\frac{g l}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} l + \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Simplify

\frac{g l}{l} : g

\frac{g l}{l}

Cancel the common factor:

l

= g

Simplify

\frac{arccos ^{2} ( - θt )}{t ^{2}} l + \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l : \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{arccos ^{2} ( - θt )}{t ^{2}} l + \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Multiply

\frac{arccos ^{2} ( - θt )}{t ^{2}} l : \frac{l arccos ^{2} ( - θt )}{t ^{2}}

\frac{arccos ^{2} ( - θt )}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{arccos ^{2} ( - θt ) l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} + l \frac{4 πn arccos ( - θt )}{t ^{2}} + l \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply

\frac{4 πn arccos ( - θt )}{t ^{2}} l : \frac{4 π l n arccos ( - θt )}{t ^{2}}

\frac{4 πn arccos ( - θt )}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 πn arccos ( - θt ) l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + l \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply

\frac{4 π ^{2} n ^{2}}{t ^{2}} l : \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{4 π ^{2} n ^{2}}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 π ^{2} n ^{2} l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

Solve

\frac{g}{l} t = - arccos (- θt) + 2 πn : g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{g}{l} t = - arccos (- θt) + 2 πn

Divide both sides by

t

\frac{g}{l} t = - arccos (- θt) + 2 πn

Divide both sides by

t

\frac{\frac{g}{l} t}{t} = - \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

Simplify

\frac{g}{l} = - \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

\frac{g}{l} = - \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

Square both sides:

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = - \frac{arccos ( - θt )}{t} + \frac{2 πn}{t}

(\frac{g}{l})^{2} = (- \frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2}

Expand

(\frac{g}{l})^{2} : \frac{g}{l}

(\frac{g}{l})^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((\frac{g}{l})^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (\frac{g}{l})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= \frac{g}{l}

Expand

(- \frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2} : \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

(- \frac{arccos ( - θt )}{t} + \frac{2 πn}{t})^{2}

Combine the fractions

- \frac{arccos ( - θt )}{t} + \frac{2 πn}{t} : \frac{- arccos ( - θt ) + 2 πn}{t}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- arccos ( - θt ) + 2 πn}{t}

= (\frac{- arccos ( - θt ) + 2 πn}{t})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( - arccos ( - θt ) + 2 πn ) ^{2}}{t ^{2}}

(- arccos (- θt) + 2 πn)^{2} = arccos^{2} (- θt) - 4 πn arccos (- θt) + 4 π^{2} n^{2}

(- arccos (- θt) + 2 πn)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - arccos (- θt), b = 2 πn

= (- arccos (- θt))^{2} + 2 (- arccos (- θt)) \cdot 2 πn + (2 πn)^{2}

Simplify

(- arccos (- θt))^{2} + 2 (- arccos (- θt)) \cdot 2 πn + (2 πn)^{2} : arccos^{2} (- θt) - 4 πn arccos (- θt) + 4 π^{2} n^{2}

(- arccos (- θt))^{2} + 2 (- arccos (- θt)) \cdot 2 πn + (2 πn)^{2}

Remove parentheses:

(- a) = - a

= (- arccos (- θt))^{2} - 2 arccos (- θt) \cdot 2 πn + (2 πn)^{2}

(- arccos (- θt))^{2} = arccos^{2} (- θt)

(- arccos (- θt))^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- arccos (- θt))^{2} = arccos^{2} (- θt)

= arccos^{2} (- θt)

2 arccos (- θt) \cdot 2 πn = 4 πn arccos (- θt)

2 arccos (- θt) \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn arccos (- θt)

(2 πn)^{2} = 4 π^{2} n^{2}

(2 πn)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2} n^{2}

2^{2} = 4

= 4 π^{2} n^{2}

= arccos^{2} (- θt) - 4 πn arccos (- θt) + 4 π^{2} n^{2}

= arccos^{2} (- θt) - 4 πn arccos (- θt) + 4 π^{2} n^{2}

= \frac{arccos ^{2} ( - θt ) - 4 πn arccos ( - θt ) + 4 π ^{2} n ^{2}}{t ^{2}}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{arccos ^{2} ( - θt ) - 4 πn arccos ( - θt ) + 4 π ^{2} n ^{2}}{t ^{2}} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

= \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Solve

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}} : g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply both sides by

l

\frac{g}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 πn arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply both sides by

l

\frac{g l}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} l - \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Simplify

\frac{g l}{l} = \frac{arccos ^{2} ( - θt )}{t ^{2}} l - \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Simplify

\frac{g l}{l} : g

\frac{g l}{l}

Cancel the common factor:

l

= g

Simplify

\frac{arccos ^{2} ( - θt )}{t ^{2}} l - \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l : \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{arccos ^{2} ( - θt )}{t ^{2}} l - \frac{4 πn arccos ( - θt )}{t ^{2}} l + \frac{4 π ^{2} n ^{2}}{t ^{2}} l

Multiply

\frac{arccos ^{2} ( - θt )}{t ^{2}} l : \frac{l arccos ^{2} ( - θt )}{t ^{2}}

\frac{arccos ^{2} ( - θt )}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{arccos ^{2} ( - θt ) l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} - l \frac{4 πn arccos ( - θt )}{t ^{2}} + l \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply

\frac{4 πn arccos ( - θt )}{t ^{2}} l : \frac{4 π l n arccos ( - θt )}{t ^{2}}

\frac{4 πn arccos ( - θt )}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 πn arccos ( - θt ) l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + l \frac{4 π ^{2} n ^{2}}{t ^{2}}

Multiply

\frac{4 π ^{2} n ^{2}}{t ^{2}} l : \frac{4 π ^{2} l n ^{2}}{t ^{2}}

\frac{4 π ^{2} n ^{2}}{t ^{2}} l

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 π ^{2} n ^{2} l}{t ^{2}}

= \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} + \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}, g = \frac{l arccos ^{2} ( - θt )}{t ^{2}} - \frac{4 π l n arccos ( - θt )}{t ^{2}} + \frac{4 π ^{2} l n ^{2}}{t ^{2}}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor g,θ(t)=-1cos(sqrt(g/l)t) ?
The general solution for solvefor g,θ(t)=-1cos(sqrt(g/l)t) is