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Popular Trigonometry >

cos(2x+60)=cos(x)

Solution

cos (2 x + 60) = cos (x)

Solution

x = \frac{4 πn}{3} - 20, x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}, x = 4 πn - 60, x = 2 π + 4 πn - 60

Degrees

x = - 1145.91559 \dots^{\circ} + 24 0^{\circ} n, x = - 1025.91559 \dots^{\circ} + 24 0^{\circ} n, x = - 3437.74677 \dots^{\circ} + 72 0^{\circ} n, x = - 3077.74677 \dots^{\circ} + 72 0^{\circ} n

Solution steps

cos (2 x + 60) = cos (x)

Subtract

cos (x)

from both sides

cos (2 x + 60) - cos (x) = 0

Rewrite using trig identities

cos (60 + 2 x) - cos (x)

Use the Sum to Product identity:

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

= - 2 sin (\frac{60 + 2 x + x}{2}) sin (\frac{60 + 2 x - x}{2})

Simplify

- 2 sin (\frac{60 + 2 x + x}{2}) sin (\frac{60 + 2 x - x}{2}) : - 2 sin (\frac{60 + 3 x}{2}) sin (\frac{60 + x}{2})

- 2 sin (\frac{60 + 2 x + x}{2}) sin (\frac{60 + 2 x - x}{2})

Add similar elements:

2 x + x = 3 x

= - 2 sin (\frac{3 x + 60}{2}) sin (\frac{2 x - x + 60}{2})

Add similar elements:

2 x - x = x

= - 2 sin (\frac{3 x + 60}{2}) sin (\frac{x + 60}{2})

= - 2 sin (\frac{60 + 3 x}{2}) sin (\frac{60 + x}{2})

- 2 sin (\frac{60 + 3 x}{2}) sin (\frac{60 + x}{2}) = 0

Solving each part separately

sin (\frac{60 + 3 x}{2}) = 0 or sin (\frac{60 + x}{2}) = 0

sin (\frac{60 + 3 x}{2}) = 0 : x = \frac{4 πn}{3} - 20, x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

sin (\frac{60 + 3 x}{2}) = 0

General solutions for

sin (\frac{60 + 3 x}{2}) = 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{60 + 3 x}{2} = 0 + 2 πn, \frac{60 + 3 x}{2} = π + 2 πn

\frac{60 + 3 x}{2} = 0 + 2 πn, \frac{60 + 3 x}{2} = π + 2 πn

Solve

\frac{60 + 3 x}{2} = 0 + 2 πn : x = \frac{4 πn}{3} - 20

\frac{60 + 3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{60 + 3 x}{2} = 2 πn

Multiply both sides by

2

\frac{60 + 3 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 60 + 3 x )}{2} = 2 \cdot 2 πn

Simplify

60 + 3 x = 4 πn

60 + 3 x = 4 πn

Move

60

to the right side

60 + 3 x = 4 πn

Subtract

60

from both sides

60 + 3 x - 60 = 4 πn - 60

Simplify

3 x = 4 πn - 60

3 x = 4 πn - 60

Divide both sides by

3

3 x = 4 πn - 60

Divide both sides by

3

\frac{3 x}{3} = \frac{4 πn}{3} - \frac{60}{3}

Simplify

x = \frac{4 πn}{3} - 20

x = \frac{4 πn}{3} - 20

Solve

\frac{60 + 3 x}{2} = π + 2 πn : x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

\frac{60 + 3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{60 + 3 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 60 + 3 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

60 + 3 x = 2 π + 4 πn

60 + 3 x = 2 π + 4 πn

Move

60

to the right side

60 + 3 x = 2 π + 4 πn

Subtract

60

from both sides

60 + 3 x - 60 = 2 π + 4 πn - 60

Simplify

3 x = 2 π + 4 πn - 60

3 x = 2 π + 4 πn - 60

Divide both sides by

3

3 x = 2 π + 4 πn - 60

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3} - \frac{60}{3}

Simplify

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3} - \frac{60}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 π}{3} + \frac{4 πn}{3} - \frac{60}{3} : \frac{2 π}{3} - 20 + \frac{4 πn}{3}

\frac{2 π}{3} + \frac{4 πn}{3} - \frac{60}{3}

Group like terms

= \frac{2 π}{3} - \frac{60}{3} + \frac{4 πn}{3}

Divide the numbers:

\frac{60}{3} = 20

= \frac{2 π}{3} - 20 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

x = \frac{4 πn}{3} - 20, x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}

sin (\frac{60 + x}{2}) = 0 : x = 4 πn - 60, x = 2 π + 4 πn - 60

sin (\frac{60 + x}{2}) = 0

General solutions for

sin (\frac{60 + x}{2}) = 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{60 + x}{2} = 0 + 2 πn, \frac{60 + x}{2} = π + 2 πn

\frac{60 + x}{2} = 0 + 2 πn, \frac{60 + x}{2} = π + 2 πn

Solve

\frac{60 + x}{2} = 0 + 2 πn : x = 4 πn - 60

\frac{60 + x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{60 + x}{2} = 2 πn

Multiply both sides by

2

\frac{60 + x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 60 + x )}{2} = 2 \cdot 2 πn

Simplify

60 + x = 4 πn

60 + x = 4 πn

Move

60

to the right side

60 + x = 4 πn

Subtract

60

from both sides

60 + x - 60 = 4 πn - 60

Simplify

x = 4 πn - 60

x = 4 πn - 60

Solve

\frac{60 + x}{2} = π + 2 πn : x = 2 π + 4 πn - 60

\frac{60 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{60 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 60 + x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

60 + x = 2 π + 4 πn

60 + x = 2 π + 4 πn

Move

60

to the right side

60 + x = 2 π + 4 πn

Subtract

60

from both sides

60 + x - 60 = 2 π + 4 πn - 60

Simplify

x = 2 π + 4 πn - 60

x = 2 π + 4 πn - 60

x = 4 πn - 60, x = 2 π + 4 πn - 60

Combine all the solutions

x = \frac{4 πn}{3} - 20, x = \frac{2 π}{3} - 20 + \frac{4 πn}{3}, x = 4 πn - 60, x = 2 π + 4 πn - 60

Graph

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