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Popular >

solvefor t,x=1-2cos(2t)

Solution

solve for

t, x = 1 - 2 cos (2 t)

Solution

t = \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn, t = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

Solution steps

x = 1 - 2 cos (2 t)

Switch sides

1 - 2 cos (2 t) = x

Move

1

to the right side

1 - 2 cos (2 t) = x

Subtract

1

from both sides

1 - 2 cos (2 t) - 1 = x - 1

Simplify

- 2 cos (2 t) = x - 1

- 2 cos (2 t) = x - 1

Divide both sides by

- 2

- 2 cos (2 t) = x - 1

Divide both sides by

- 2

\frac{- 2 cos ( 2 t )}{- 2} = \frac{x}{- 2} - \frac{1}{- 2}

Simplify

\frac{- 2 cos ( 2 t )}{- 2} = \frac{x}{- 2} - \frac{1}{- 2}

Simplify

\frac{- 2 cos ( 2 t )}{- 2} : cos (2 t)

\frac{- 2 cos ( 2 t )}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2 cos ( 2 t )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos (2 t)

Simplify

\frac{x}{- 2} - \frac{1}{- 2} : - \frac{x - 1}{2}

\frac{x}{- 2} - \frac{1}{- 2}

Apply rule

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

= \frac{x - 1}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{x - 1}{2}

cos (2 t) = - \frac{x - 1}{2}

cos (2 t) = - \frac{x - 1}{2}

cos (2 t) = - \frac{x - 1}{2}

Apply trig inverse properties

cos (2 t) = - \frac{x - 1}{2}

General solutions for

cos (2 t) = - \frac{x - 1}{2}

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

2 t = arccos (- \frac{x - 1}{2}) + 2 πn, 2 t = - arccos (- \frac{x - 1}{2}) + 2 πn

2 t = arccos (- \frac{x - 1}{2}) + 2 πn, 2 t = - arccos (- \frac{x - 1}{2}) + 2 πn

Solve

2 t = arccos (- \frac{x - 1}{2}) + 2 πn : t = \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

2 t = arccos (- \frac{x - 1}{2}) + 2 πn

Divide both sides by

2

2 t = arccos (- \frac{x - 1}{2}) + 2 πn

Divide both sides by

2

\frac{2 t}{2} = \frac{arccos ( - \frac{x - 1}{2} )}{2} + \frac{2 πn}{2}

Simplify

t = \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

t = \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

Solve

2 t = - arccos (- \frac{x - 1}{2}) + 2 πn : t = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

2 t = - arccos (- \frac{x - 1}{2}) + 2 πn

Divide both sides by

2

2 t = - arccos (- \frac{x - 1}{2}) + 2 πn

Divide both sides by

2

\frac{2 t}{2} = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + \frac{2 πn}{2}

Simplify

t = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

t = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

t = \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn, t = - \frac{arccos ( - \frac{x - 1}{2} )}{2} + πn

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