What is the general solution for cos(x)=sin(x-pi/3) ?

The general solution for cos(x)=sin(x-pi/3) is x=1.30899…+pin

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cos(x)=sin(x-pi/3)

Solution

cos (x) = sin (x - \frac{π}{3})

Solution

x = 1.30899 \dots + πn

Degrees

x = 7 5^{\circ} + 18 0^{\circ} n

Solution steps

cos (x) = sin (x - \frac{π}{3})

Rewrite using trig identities

cos (x) = sin (x - \frac{π}{3})

Rewrite using trig identities

sin (x - \frac{π}{3})

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

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

Simplify

sin (x) cos (\frac{π}{3}) - cos (x) sin (\frac{π}{3}) : \frac{1}{2} sin (x) - \frac{3}{2} cos (x)

sin (x) cos (\frac{π}{3}) - cos (x) sin (\frac{π}{3})

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

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}

= \frac{1}{2}

= \frac{1}{2} sin (x) - sin (\frac{π}{3}) cos (x)

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{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{3}{2}

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

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

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

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

Subtract

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

from both sides

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

Simplify

- \frac{1}{2} sin (x) + \frac{3 + 2}{2} cos (x) : \frac{- sin ( x ) + ( 3 + 2 ) cos ( x )}{2}

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

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiply fractions:

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

= \frac{1 \cdot sin ( x )}{2}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{2}

\frac{3 + 2}{2} cos (x) = \frac{( 3 + 2 ) cos ( x )}{2}

\frac{3 + 2}{2} cos (x)

Multiply fractions:

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

= \frac{( 3 + 2 ) cos ( x )}{2}

= - \frac{sin ( x )}{2} + \frac{( 2 + 3 ) cos ( x )}{2}

Apply rule

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

= \frac{- sin ( x ) + ( 2 + 3 ) cos ( x )}{2}

\frac{- sin ( x ) + ( 3 + 2 ) cos ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- sin (x) + (3 + 2) cos (x) = 0

Rewrite using trig identities

- sin (x) + (3 + 2) cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{- sin ( x ) + ( 3 + 2 ) cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- \frac{sin ( x )}{cos ( x )} + 3 + 2 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- tan (x) + 3 + 2 = 0

- tan (x) + 3 + 2 = 0

Move

3

to the right side

- tan (x) + 3 + 2 = 0

Subtract

3

from both sides

- tan (x) + 3 + 2 - 3 = 0 - 3

Simplify

- tan (x) + 2 = - 3

- tan (x) + 2 = - 3

Move

2

to the right side

- tan (x) + 2 = - 3

Subtract

2

from both sides

- tan (x) + 2 - 2 = - 3 - 2

Simplify

- tan (x) = - 3 - 2

- tan (x) = - 3 - 2

Divide both sides by

- 1

- tan (x) = - 3 - 2

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = - \frac{3}{- 1} - \frac{2}{- 1}

Simplify

\frac{- tan ( x )}{- 1} = - \frac{3}{- 1} - \frac{2}{- 1}

Simplify

\frac{- tan ( x )}{- 1} : tan (x)

\frac{- tan ( x )}{- 1}

Apply the fraction rule:

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

= \frac{tan ( x )}{1}

Apply rule

\frac{a}{1} = a

= tan (x)

Simplify

- \frac{3}{- 1} - \frac{2}{- 1} : 2 + 3

- \frac{3}{- 1} - \frac{2}{- 1}

Apply rule

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

= \frac{- 3 - 2}{- 1}

Apply the fraction rule:

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

- 3 - 2 = - (2 + 3)

= \frac{2 + 3}{1}

Apply rule

\frac{a}{1} = a

= 2 + 3

tan (x) = 2 + 3

tan (x) = 2 + 3

tan (x) = 2 + 3

Apply trig inverse properties

tan (x) = 2 + 3

General solutions for

tan (x) = 2 + 3

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2 + 3) + πn

x = arctan (2 + 3) + πn

Show solutions in decimal form

x = 1.30899 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)=sin(x-pi/3) ?
The general solution for cos(x)=sin(x-pi/3) is x=1.30899…+pin