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solvefor t,f=acos((tpi)/6-(11pi)/(12))

Solution

solve for

t, f = a cos (\frac{t π}{6} - \frac{11 π}{12})

Solution

t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}, t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

Solution steps

f = a cos (\frac{t π}{6} - \frac{11 π}{12})

Switch sides

a cos (\frac{t π}{6} - \frac{11 π}{12}) = f

Divide both sides by

a; a \neq = 0

a cos (\frac{t π}{6} - \frac{11 π}{12}) = f

Divide both sides by

a; a \neq = 0

\frac{a cos ( \frac{t π}{6} - \frac{11 π}{12} )}{a} = \frac{f}{a}; a \neq = 0

Simplify

cos (\frac{t π}{6} - \frac{11 π}{12}) = \frac{f}{a}; a \neq = 0

cos (\frac{t π}{6} - \frac{11 π}{12}) = \frac{f}{a}; a \neq = 0

Apply trig inverse properties

cos (\frac{t π}{6} - \frac{11 π}{12}) = \frac{f}{a}

General solutions for

cos (\frac{t π}{6} - \frac{11 π}{12}) = \frac{f}{a}

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

\frac{t π}{6} - \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn, \frac{t π}{6} - \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn

\frac{t π}{6} - \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn, \frac{t π}{6} - \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn

Solve

\frac{t π}{6} - \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn : t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

\frac{t π}{6} - \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn

Move

\frac{11 π}{12}

to the right side

\frac{t π}{6} - \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn

Add

\frac{11 π}{12}

to both sides

\frac{t π}{6} - \frac{11 π}{12} + \frac{11 π}{12} = arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Simplify

\frac{t π}{6} = arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

\frac{t π}{6} = arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Multiply both sides by

6

\frac{t π}{6} = arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Multiply both sides by

6

\frac{6 t π}{6} = 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

Simplify

\frac{6 t π}{6} = 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

Simplify

\frac{6 t π}{6} : π t

\frac{6 t π}{6}

Divide the numbers:

\frac{6}{6} = 1

= π t

Simplify

6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12} : 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

6 \cdot \frac{11 π}{12} = \frac{11 π}{2}

6 \cdot \frac{11 π}{12}

Multiply fractions:

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

= \frac{11 π 6}{12}

Multiply the numbers:

11 \cdot 6 = 66

= \frac{66 π}{12}

Cancel the common factor:

6

= \frac{11 π}{2}

= 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

Divide both sides by

π

π t = 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

Divide both sides by

π

\frac{π t}{π} = \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

Simplify

\frac{π t}{π} = \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π} : \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

\frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

\frac{12 πn}{π} = 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

\frac{\frac{11 π}{2}}{π} = \frac{11}{2}

\frac{\frac{11 π}{2}}{π}

Apply the fraction rule:

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

= \frac{11 π}{2 π}

Cancel the common factor:

π

= \frac{11}{2}

= \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

Solve

\frac{t π}{6} - \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn : t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

\frac{t π}{6} - \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn

Move

\frac{11 π}{12}

to the right side

\frac{t π}{6} - \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn

Add

\frac{11 π}{12}

to both sides

\frac{t π}{6} - \frac{11 π}{12} + \frac{11 π}{12} = - arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Simplify

\frac{t π}{6} = - arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

\frac{t π}{6} = - arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Multiply both sides by

6

\frac{t π}{6} = - arccos (\frac{f}{a}) + 2 πn + \frac{11 π}{12}

Multiply both sides by

6

\frac{6 t π}{6} = - 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

Simplify

\frac{6 t π}{6} = - 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

Simplify

\frac{6 t π}{6} : π t

\frac{6 t π}{6}

Divide the numbers:

\frac{6}{6} = 1

= π t

Simplify

- 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12} : - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

- 6 arccos (\frac{f}{a}) + 6 \cdot 2 πn + 6 \cdot \frac{11 π}{12}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

6 \cdot \frac{11 π}{12} = \frac{11 π}{2}

6 \cdot \frac{11 π}{12}

Multiply fractions:

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

= \frac{11 π 6}{12}

Multiply the numbers:

11 \cdot 6 = 66

= \frac{66 π}{12}

Cancel the common factor:

6

= \frac{11 π}{2}

= - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

π t = - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

Divide both sides by

π

π t = - 6 arccos (\frac{f}{a}) + 12 πn + \frac{11 π}{2}

Divide both sides by

π

\frac{π t}{π} = - \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

Simplify

\frac{π t}{π} = - \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

- \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π} : - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

- \frac{6 arccos ( \frac{f}{a} )}{π} + \frac{12 πn}{π} + \frac{\frac{11 π}{2}}{π}

\frac{12 πn}{π} = 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

\frac{\frac{11 π}{2}}{π} = \frac{11}{2}

\frac{\frac{11 π}{2}}{π}

Apply the fraction rule:

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

= \frac{11 π}{2 π}

Cancel the common factor:

π

= \frac{11}{2}

= - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

t = \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}, t = - \frac{6 arccos ( \frac{f}{a} )}{π} + 12 n + \frac{11}{2}

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