What is the general solution for 3cos((2pit)/3)+10=11 ?

The general solution for 3cos((2pit)/3)+10=11 is t=(3*1.23095…)/(2pi)+3n,t=3-(3*1.23095…)/(2pi)+3n

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Popular Trigonometry >

3cos((2pit)/3)+10=11

Solution

3 cos (\frac{2 π t}{3}) + 10 = 11

Solution

t = \frac{3 \cdot 1.23095 \dots}{2 π} + 3 n, t = 3 - \frac{3 \cdot 1.23095 \dots}{2 π} + 3 n

Degrees

t = 33.67501 \dots^{\circ} + 171.88733 \dots^{\circ} n, t = 138.21232 \dots^{\circ} + 171.88733 \dots^{\circ} n

Solution steps

3 cos (\frac{2 π t}{3}) + 10 = 11

Move

10

to the right side

3 cos (\frac{2 π t}{3}) + 10 = 11

Subtract

10

from both sides

3 cos (\frac{2 π t}{3}) + 10 - 10 = 11 - 10

Simplify

3 cos (\frac{2 π t}{3}) = 1

3 cos (\frac{2 π t}{3}) = 1

Divide both sides by

3

3 cos (\frac{2 π t}{3}) = 1

Divide both sides by

3

\frac{3 cos ( \frac{2 π t}{3} )}{3} = \frac{1}{3}

Simplify

cos (\frac{2 π t}{3}) = \frac{1}{3}

cos (\frac{2 π t}{3}) = \frac{1}{3}

Apply trig inverse properties

cos (\frac{2 π t}{3}) = \frac{1}{3}

General solutions for

cos (\frac{2 π t}{3}) = \frac{1}{3}

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

\frac{2 π t}{3} = arccos (\frac{1}{3}) + 2 πn, \frac{2 π t}{3} = 2 π - arccos (\frac{1}{3}) + 2 πn

\frac{2 π t}{3} = arccos (\frac{1}{3}) + 2 πn, \frac{2 π t}{3} = 2 π - arccos (\frac{1}{3}) + 2 πn

Solve

\frac{2 π t}{3} = arccos (\frac{1}{3}) + 2 πn : t = \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

\frac{2 π t}{3} = arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

3

\frac{2 π t}{3} = arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 π t}{3} = 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π t}{3} = 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π t}{3} : 2 π t

\frac{3 \cdot 2 π t}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π t}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π t

Simplify

3 arccos (\frac{1}{3}) + 3 \cdot 2 πn : 3 arccos (\frac{1}{3}) + 6 πn

3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 3 arccos (\frac{1}{3}) + 6 πn

Divide both sides by

2 π

2 π t = 3 arccos (\frac{1}{3}) + 6 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π} : \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

\frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

Solve

\frac{2 π t}{3} = 2 π - arccos (\frac{1}{3}) + 2 πn : t = 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

\frac{2 π t}{3} = 2 π - arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

3

\frac{2 π t}{3} = 2 π - arccos (\frac{1}{3}) + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 π t}{3} = 3 \cdot 2 π - 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π t}{3} = 3 \cdot 2 π - 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π t}{3} : 2 π t

\frac{3 \cdot 2 π t}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π t}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π t

Simplify

3 \cdot 2 π - 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn : 6 π - 3 arccos (\frac{1}{3}) + 6 πn

3 \cdot 2 π - 3 arccos (\frac{1}{3}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 π - 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 6 π - 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 6 π - 3 arccos (\frac{1}{3}) + 6 πn

2 π t = 6 π - 3 arccos (\frac{1}{3}) + 6 πn

Divide both sides by

2 π

2 π t = 6 π - 3 arccos (\frac{1}{3}) + 6 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{6 π}{2 π} - \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{6 π}{2 π} - \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π t}{2 π} : t

\frac{2 π t}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{6 π}{2 π} - \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π} : 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

\frac{6 π}{2 π} - \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 π}{2 π} : 3

\frac{6 π}{2 π}

Cancel

\frac{6 π}{2 π} : 3

\frac{6 π}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3

= 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

t = \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n, t = 3 - \frac{3 arccos ( \frac{1}{3} )}{2 π} + 3 n

Show solutions in decimal form

t = \frac{3 \cdot 1.23095 \dots}{2 π} + 3 n, t = 3 - \frac{3 \cdot 1.23095 \dots}{2 π} + 3 n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos((2pit)/3)+10=11 ?
The general solution for 3cos((2pit)/3)+10=11 is t=(3*1.23095…)/(2pi)+3n,t=3-(3*1.23095…)/(2pi)+3n