What is the general solution for cos(pi/6 t)=cos(pi/6 (t-3)) ?

The general solution for cos(pi/6 t)=cos(pi/6 (t-3)) is t=12n+3/2 ,t= 15/2+12n

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cos(pi/6 t)=cos(pi/6 (t-3))

Solution

cos (\frac{π}{6} t) = cos (\frac{π}{6} (t - 3))

Solution

t = 12 n + \frac{3}{2}, t = \frac{15}{2} + 12 n

Degrees

t = 85.94366 \dots^{\circ} + 687.54935 \dots^{\circ} n, t = 429.71834 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

cos (\frac{π}{6} t) = cos (\frac{π}{6} (t - 3))

Subtract

cos (\frac{π}{6} (t - 3))

from both sides

cos (\frac{π t}{6}) - cos (\frac{π}{6} (t - 3)) = 0

Simplify

cos (\frac{π t}{6}) - cos (\frac{π}{6} (t - 3)) : cos (\frac{π t}{6}) - cos (\frac{π ( t - 3 )}{6})

cos (\frac{π t}{6}) - cos (\frac{π}{6} (t - 3))

Multiply

\frac{π}{6} (t - 3) : \frac{π ( t - 3 )}{6}

\frac{π}{6} (t - 3)

Multiply fractions:

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

= \frac{π ( t - 3 )}{6}

= cos (\frac{π t}{6}) - cos (\frac{π ( t - 3 )}{6})

cos (\frac{π t}{6}) - cos (\frac{π ( t - 3 )}{6}) = 0

Rewrite using trig identities

cos (\frac{π t}{6}) - cos (\frac{π ( t - 3 )}{6})

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

Simplify

- 2 sin (\frac{\frac{π t}{6} + \frac{π ( t - 3 )}{6}}{2}) sin (\frac{\frac{π t}{6} - \frac{π ( t - 3 )}{6}}{2}) : - 2 sin (\frac{2 π t - 3 π}{12})

- 2 sin (\frac{\frac{π t}{6} + \frac{π ( t - 3 )}{6}}{2}) sin (\frac{\frac{π t}{6} - \frac{π ( t - 3 )}{6}}{2})

\frac{\frac{π t}{6} + \frac{π ( t - 3 )}{6}}{2} = \frac{2 π t - 3 π}{12}

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

Combine the fractions

\frac{π t}{6} + \frac{π ( t - 3 )}{6} : \frac{π t + π ( t - 3 )}{6}

Apply rule

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

= \frac{π t + π ( t - 3 )}{6}

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

Apply the fraction rule:

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

= \frac{π t + π ( t - 3 )}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π t + π ( t - 3 )}{12}

Expand

π t + π (t - 3) : 2 π t - 3 π

π t + π (t - 3)

Expand

π (t - 3) : π t - 3 π

π (t - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = π, b = t, c = 3

= π t - π 3

= π t - 3 π

= π t + π t - 3 π

Add similar elements:

π t + π t = 2 π t

= 2 π t - 3 π

= \frac{2 π t - 3 π}{12}

= - 2 sin (\frac{2 π t - 3 π}{12}) sin (\frac{\frac{π t}{6} - \frac{π ( t - 3 )}{6}}{2})

\frac{\frac{π t}{6} - \frac{π ( t - 3 )}{6}}{2} = \frac{π}{4}

\frac{\frac{π t}{6} - \frac{π ( t - 3 )}{6}}{2}

Combine the fractions

\frac{π t}{6} - \frac{π ( t - 3 )}{6} : \frac{π}{2}

Apply rule

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

= \frac{π t - π ( t - 3 )}{6}

Factor

π t - π (t - 3) : π 3

π t - π (t - 3)

Factor out common term

π

= π (t - (- 3 + t))

Expand

t - (t - 3) : 3

t - (- 3 + t)

- (- 3 + t) : 3 - t

- (- 3 + t)

Distribute parentheses

= - (- 3) - (t)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 3 - t

= t + 3 - t

Simplify

t + 3 - t : 3

t + 3 - t

Group like terms

= t - t + 3

Add similar elements:

t - t = 0

= 3

= 3

= 3 π

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= - 2 sin (\frac{π}{4}) sin (\frac{2 π t - 3 π}{12})

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= - 2 \cdot \frac{2}{2} sin (\frac{2 π t - 3 π}{12})

Multiply fractions:

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

= - \frac{2 \cdot 2 sin ( \frac{2 π t - 3 π}{12} )}{2}

Cancel the common factor:

2

= - 2 sin (\frac{2 π t - 3 π}{12})

= - 2 sin (\frac{2 π t - 3 π}{12})

- 2 sin (\frac{2 π t - 3 π}{12}) = 0

Divide both sides by

- 2

- 2 sin (\frac{2 π t - 3 π}{12}) = 0

Divide both sides by

- 2

\frac{- 2 sin ( \frac{2 π t - 3 π}{12} )}{- 2} = \frac{0}{- 2}

Simplify

sin (\frac{2 π t - 3 π}{12}) = 0

sin (\frac{2 π t - 3 π}{12}) = 0

General solutions for

sin (\frac{2 π t - 3 π}{12}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{2 π t - 3 π}{12} = 0 + 2 πn, \frac{2 π t - 3 π}{12} = π + 2 πn

\frac{2 π t - 3 π}{12} = 0 + 2 πn, \frac{2 π t - 3 π}{12} = π + 2 πn

Solve

\frac{2 π t - 3 π}{12} = 0 + 2 πn : t = 12 n + \frac{3}{2}

\frac{2 π t - 3 π}{12} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{2 π t - 3 π}{12} = 2 πn

Multiply both sides by

12

\frac{2 π t - 3 π}{12} = 2 πn

Multiply both sides by

12

\frac{12 ( 2 π t - 3 π )}{12} = 12 \cdot 2 πn

Simplify

2 π t - 3 π = 24 πn

2 π t - 3 π = 24 πn

Move

3 π

to the right side

2 π t - 3 π = 24 πn

Add

3 π

to both sides

2 π t - 3 π + 3 π = 24 πn + 3 π

Simplify

2 π t = 24 πn + 3 π

2 π t = 24 πn + 3 π

Divide both sides by

2 π

2 π t = 24 πn + 3 π

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{24 πn}{2 π} + \frac{3 π}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{24 πn}{2 π} + \frac{3 π}{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{24 πn}{2 π} + \frac{3 π}{2 π} : 12 n + \frac{3}{2}

\frac{24 πn}{2 π} + \frac{3 π}{2 π}

Cancel

\frac{24 πn}{2 π} : 12 n

\frac{24 πn}{2 π}

Cancel

\frac{24 πn}{2 π} : 12 n

\frac{24 πn}{2 π}

Divide the numbers:

\frac{24}{2} = 12

= \frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 12 n

= 12 n + \frac{3 π}{2 π}

Cancel

\frac{3 π}{2 π} : \frac{3}{2}

\frac{3 π}{2 π}

Cancel the common factor:

π

= \frac{3}{2}

= 12 n + \frac{3}{2}

t = 12 n + \frac{3}{2}

t = 12 n + \frac{3}{2}

t = 12 n + \frac{3}{2}

Solve

\frac{2 π t - 3 π}{12} = π + 2 πn : t = \frac{15}{2} + 12 n

\frac{2 π t - 3 π}{12} = π + 2 πn

Multiply both sides by

12

\frac{2 π t - 3 π}{12} = π + 2 πn

Multiply both sides by

12

\frac{12 ( 2 π t - 3 π )}{12} = 12 π + 12 \cdot 2 πn

Simplify

2 π t - 3 π = 12 π + 24 πn

2 π t - 3 π = 12 π + 24 πn

Move

3 π

to the right side

2 π t - 3 π = 12 π + 24 πn

Add

3 π

to both sides

2 π t - 3 π + 3 π = 12 π + 24 πn + 3 π

Simplify

2 π t = 15 π + 24 πn

2 π t = 15 π + 24 πn

Divide both sides by

2 π

2 π t = 15 π + 24 πn

Divide both sides by

2 π

\frac{2 π t}{2 π} = \frac{15 π}{2 π} + \frac{24 πn}{2 π}

Simplify

\frac{2 π t}{2 π} = \frac{15 π}{2 π} + \frac{24 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{15 π}{2 π} + \frac{24 πn}{2 π} : \frac{15}{2} + 12 n

\frac{15 π}{2 π} + \frac{24 πn}{2 π}

Cancel

\frac{15 π}{2 π} : \frac{15}{2}

\frac{15 π}{2 π}

Cancel the common factor:

π

= \frac{15}{2}

= \frac{15}{2} + \frac{24 πn}{2 π}

Cancel

\frac{24 πn}{2 π} : 12 n

\frac{24 πn}{2 π}

Cancel

\frac{24 πn}{2 π} : 12 n

\frac{24 πn}{2 π}

Divide the numbers:

\frac{24}{2} = 12

= \frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 12 n

= \frac{15}{2} + 12 n

t = \frac{15}{2} + 12 n

t = \frac{15}{2} + 12 n

t = \frac{15}{2} + 12 n

t = 12 n + \frac{3}{2}, t = \frac{15}{2} + 12 n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/6 t)=cos(pi/6 (t-3)) ?
The general solution for cos(pi/6 t)=cos(pi/6 (t-3)) is t=12n+3/2 ,t= 15/2+12n