What is the general solution for 0=cos(2t-pi/3) ?

The general solution for 0=cos(2t-pi/3) is t=pin+(5pi)/(12),t=pin+(11pi)/(12)

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Popular >

0=cos(2t-pi/3)

Solution

0 = cos (2 t - \frac{π}{3})

Solution

t = πn + \frac{5 π}{12}, t = πn + \frac{11 π}{12}

Degrees

t = 7 5^{\circ} + 18 0^{\circ} n, t = 16 5^{\circ} + 18 0^{\circ} n

Solution steps

0 = cos (2 t - \frac{π}{3})

Switch sides

cos (2 t - \frac{π}{3}) = 0

General solutions for

cos (2 t - \frac{π}{3}) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Solve

2 t - \frac{π}{3} = \frac{π}{2} + 2 πn : t = πn + \frac{5 π}{12}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

2 t - \frac{π}{3} + \frac{π}{3} : 2 t

2 t - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 t

Simplify

\frac{π}{2} + 2 πn + \frac{π}{3} : 2 πn + \frac{5 π}{6}

\frac{π}{2} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π 3}{6} + \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π 2}{6}

Add similar elements:

3 π + 2 π = 5 π

= 2 πn + \frac{5 π}{6}

2 t = 2 πn + \frac{5 π}{6}

2 t = 2 πn + \frac{5 π}{6}

2 t = 2 πn + \frac{5 π}{6}

Divide both sides by

2

2 t = 2 πn + \frac{5 π}{6}

Divide both sides by

2

\frac{2 t}{2} = \frac{2 πn}{2} + \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 t}{2} = \frac{2 πn}{2} + \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 t}{2} : t

\frac{2 t}{2}

Divide the numbers:

\frac{2}{2} = 1

= t

Simplify

\frac{2 πn}{2} + \frac{\frac{5 π}{6}}{2} : πn + \frac{5 π}{12}

\frac{2 πn}{2} + \frac{\frac{5 π}{6}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

= πn + \frac{5 π}{12}

t = πn + \frac{5 π}{12}

t = πn + \frac{5 π}{12}

t = πn + \frac{5 π}{12}

Solve

2 t - \frac{π}{3} = \frac{3 π}{2} + 2 πn : t = πn + \frac{11 π}{12}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

2 t - \frac{π}{3} + \frac{π}{3} : 2 t

2 t - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 t

Simplify

\frac{3 π}{2} + 2 πn + \frac{π}{3} : 2 πn + \frac{11 π}{6}

\frac{3 π}{2} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{3 π}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

For

\frac{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= \frac{π 2}{6} + \frac{9 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 9 π}{6}

Add similar elements:

2 π + 9 π = 11 π

= 2 πn + \frac{11 π}{6}

2 t = 2 πn + \frac{11 π}{6}

2 t = 2 πn + \frac{11 π}{6}

2 t = 2 πn + \frac{11 π}{6}

Divide both sides by

2

2 t = 2 πn + \frac{11 π}{6}

Divide both sides by

2

\frac{2 t}{2} = \frac{2 πn}{2} + \frac{\frac{11 π}{6}}{2}

Simplify

\frac{2 t}{2} = \frac{2 πn}{2} + \frac{\frac{11 π}{6}}{2}

Simplify

\frac{2 t}{2} : t

\frac{2 t}{2}

Divide the numbers:

\frac{2}{2} = 1

= t

Simplify

\frac{2 πn}{2} + \frac{\frac{11 π}{6}}{2} : πn + \frac{11 π}{12}

\frac{2 πn}{2} + \frac{\frac{11 π}{6}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

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

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

= πn + \frac{11 π}{12}

t = πn + \frac{11 π}{12}

t = πn + \frac{11 π}{12}

t = πn + \frac{11 π}{12}

t = πn + \frac{5 π}{12}, t = πn + \frac{11 π}{12}

Popular Examples

sin(0.5x)=0

sin (0.5 x) = 0

cos(2x)=2-3sin(x), pi/2 <= x<= (3pi)/2

cos (2 x) = 2 - 3 sin (x), \frac{π}{2} \leq x \leq \frac{3 π}{2}

3sin(x)cos(x)=0

3 sin (x) cos (x) = 0

1/2 sin(t)-1/2 =-1

\frac{1}{2} sin (t) - \frac{1}{2} = - 1

20sin^2(x)-21sin(x)+4=0

20 sin^{2} (x) - 21 sin (x) + 4 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 0=cos(2t-pi/3) ?
The general solution for 0=cos(2t-pi/3) is t=pin+(5pi)/(12),t=pin+(11pi)/(12)