What is the general solution for 0=12cos(pi/3 t)+8 ?

The general solution for 0=12cos(pi/3 t)+8 is t=(3*2.30052…)/pi+6n,t=-(3*2.30052…)/pi+6n

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0=12cos(pi/3 t)+8

Solution

0 = 12 cos (\frac{π}{3} t) + 8

Solution

t = \frac{3 \cdot 2.30052 \dots}{π} + 6 n, t = - \frac{3 \cdot 2.30052 \dots}{π} + 6 n

Degrees

t = 125.86957 \dots^{\circ} + 343.77467 \dots^{\circ} n, t = - 125.86957 \dots^{\circ} + 343.77467 \dots^{\circ} n

Solution steps

0 = 12 cos (\frac{π}{3} t) + 8

Switch sides

12 cos (\frac{π}{3} t) + 8 = 0

Move

8

to the right side

12 cos (\frac{π}{3} t) + 8 = 0

Subtract

8

from both sides

12 cos (\frac{π}{3} t) + 8 - 8 = 0 - 8

Simplify

12 cos (\frac{π}{3} t) = - 8

12 cos (\frac{π}{3} t) = - 8

Divide both sides by

12

12 cos (\frac{π}{3} t) = - 8

Divide both sides by

12

\frac{12 cos ( \frac{π}{3} t )}{12} = \frac{- 8}{12}

Simplify

\frac{12 cos ( \frac{π}{3} t )}{12} = \frac{- 8}{12}

Simplify

\frac{12 cos ( \frac{π}{3} t )}{12} : cos (\frac{π}{3} t)

\frac{12 cos ( \frac{π}{3} t )}{12}

Divide the numbers:

\frac{12}{12} = 1

= cos (\frac{π}{3} t)

Simplify

\frac{- 8}{12} : - \frac{2}{3}

\frac{- 8}{12}

Apply the fraction rule:

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

= - \frac{8}{12}

Cancel the common factor:

4

= - \frac{2}{3}

cos (\frac{π}{3} t) = - \frac{2}{3}

cos (\frac{π}{3} t) = - \frac{2}{3}

cos (\frac{π}{3} t) = - \frac{2}{3}

Apply trig inverse properties

cos (\frac{π}{3} t) = - \frac{2}{3}

General solutions for

cos (\frac{π}{3} t) = - \frac{2}{3}

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

\frac{π}{3} t = arccos (- \frac{2}{3}) + 2 πn, \frac{π}{3} t = - arccos (- \frac{2}{3}) + 2 πn

\frac{π}{3} t = arccos (- \frac{2}{3}) + 2 πn, \frac{π}{3} t = - arccos (- \frac{2}{3}) + 2 πn

Solve

\frac{π}{3} t = arccos (- \frac{2}{3}) + 2 πn : t = \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

\frac{π}{3} t = arccos (- \frac{2}{3}) + 2 πn

Multiply both sides by

3

\frac{π}{3} t = arccos (- \frac{2}{3}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{π}{3} t = 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t = 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t : π t

3 \cdot \frac{π}{3} t

Multiply fractions:

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

= \frac{3 π}{3} t

Cancel the common factor:

3

= t π

Simplify

3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn : 3 arccos (- \frac{2}{3}) + 6 πn

3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arccos (- \frac{2}{3}) + 6 πn

π t = 3 arccos (- \frac{2}{3}) + 6 πn

π t = 3 arccos (- \frac{2}{3}) + 6 πn

π t = 3 arccos (- \frac{2}{3}) + 6 πn

Divide both sides by

π

π t = 3 arccos (- \frac{2}{3}) + 6 πn

Divide both sides by

π

\frac{π t}{π} = \frac{3 arccos ( - \frac{2}{3} )}{π} + \frac{6 πn}{π}

Simplify

t = \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

t = \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

Solve

\frac{π}{3} t = - arccos (- \frac{2}{3}) + 2 πn : t = - \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

\frac{π}{3} t = - arccos (- \frac{2}{3}) + 2 πn

Multiply both sides by

3

\frac{π}{3} t = - arccos (- \frac{2}{3}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{π}{3} t = - 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t = - 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t : π t

3 \cdot \frac{π}{3} t

Multiply fractions:

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

= \frac{3 π}{3} t

Cancel the common factor:

3

= t π

Simplify

- 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn : - 3 arccos (- \frac{2}{3}) + 6 πn

- 3 arccos (- \frac{2}{3}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= - 3 arccos (- \frac{2}{3}) + 6 πn

π t = - 3 arccos (- \frac{2}{3}) + 6 πn

π t = - 3 arccos (- \frac{2}{3}) + 6 πn

π t = - 3 arccos (- \frac{2}{3}) + 6 πn

Divide both sides by

π

π t = - 3 arccos (- \frac{2}{3}) + 6 πn

Divide both sides by

π

\frac{π t}{π} = - \frac{3 arccos ( - \frac{2}{3} )}{π} + \frac{6 πn}{π}

Simplify

t = - \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

t = - \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

t = \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n, t = - \frac{3 arccos ( - \frac{2}{3} )}{π} + 6 n

Show solutions in decimal form

t = \frac{3 \cdot 2.30052 \dots}{π} + 6 n, t = - \frac{3 \cdot 2.30052 \dots}{π} + 6 n

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Frequently Asked Questions (FAQ)

What is the general solution for 0=12cos(pi/3 t)+8 ?
The general solution for 0=12cos(pi/3 t)+8 is t=(3*2.30052…)/pi+6n,t=-(3*2.30052…)/pi+6n