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Popular Trigonometry >

cos(4θ)+cos(2θ)=cos(θ)

Solution

cos (4 θ) + cos (2 θ) = cos (θ)

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{9} + \frac{2 πn}{3}, θ = \frac{5 π}{9} + \frac{2 πn}{3}

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 2 0^{\circ} + 12 0^{\circ} n, θ = 10 0^{\circ} + 12 0^{\circ} n

Solution steps

cos (4 θ) + cos (2 θ) = cos (θ)

Subtract

cos (θ)

from both sides

cos (4 θ) + cos (2 θ) - cos (θ) = 0

Rewrite using trig identities

cos (2 θ) + cos (4 θ) - cos (θ)

Use the Sum to Product identity:

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

= - cos (θ) + 2 cos (\frac{2 θ + 4 θ}{2}) cos (\frac{2 θ - 4 θ}{2})

2 cos (\frac{2 θ + 4 θ}{2}) cos (\frac{2 θ - 4 θ}{2}) = 2 cos (θ) cos (3 θ)

2 cos (\frac{2 θ + 4 θ}{2}) cos (\frac{2 θ - 4 θ}{2})

\frac{2 θ + 4 θ}{2} = 3 θ

\frac{2 θ + 4 θ}{2}

Add similar elements:

2 θ + 4 θ = 6 θ

= \frac{6 θ}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 θ

= 2 cos (3 θ) cos (\frac{2 θ - 4 θ}{2})

\frac{2 θ - 4 θ}{2} = - θ

\frac{2 θ - 4 θ}{2}

Add similar elements:

2 θ - 4 θ = - 2 θ

= \frac{- 2 θ}{2}

Apply the fraction rule:

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

= - \frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= - θ

= 2 cos (3 θ) cos (- θ)

Use the negative angle identity:

cos (- x) = cos (x)

= 2 cos (θ) cos (3 θ)

= - cos (θ) + 2 cos (θ) cos (3 θ)

- cos (θ) + 2 cos (3 θ) cos (θ) = 0

Factor

- cos (θ) + 2 cos (3 θ) cos (θ) : cos (θ) (2 cos (3 θ) - 1)

- cos (θ) + 2 cos (3 θ) cos (θ)

Factor out common term

cos (θ)

= cos (θ) (- 1 + 2 cos (3 θ))

cos (θ) (2 cos (3 θ) - 1) = 0

Solving each part separately

cos (θ) = 0 or 2 cos (3 θ) - 1 = 0

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

cos (θ) = 0

General solutions for

cos (θ) = 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}

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

2 cos (3 θ) - 1 = 0 : θ = \frac{π}{9} + \frac{2 πn}{3}, θ = \frac{5 π}{9} + \frac{2 πn}{3}

2 cos (3 θ) - 1 = 0

Move

1

to the right side

2 cos (3 θ) - 1 = 0

Add

1

to both sides

2 cos (3 θ) - 1 + 1 = 0 + 1

Simplify

2 cos (3 θ) = 1

2 cos (3 θ) = 1

Divide both sides by

2

2 cos (3 θ) = 1

Divide both sides by

2

\frac{2 cos ( 3 θ )}{2} = \frac{1}{2}

Simplify

cos (3 θ) = \frac{1}{2}

cos (3 θ) = \frac{1}{2}

General solutions for

cos (3 θ) = \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}

3 θ = \frac{π}{3} + 2 πn, 3 θ = \frac{5 π}{3} + 2 πn

3 θ = \frac{π}{3} + 2 πn, 3 θ = \frac{5 π}{3} + 2 πn

Solve

3 θ = \frac{π}{3} + 2 πn : θ = \frac{π}{9} + \frac{2 πn}{3}

3 θ = \frac{π}{3} + 2 πn

Divide both sides by

3

3 θ = \frac{π}{3} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{9} + \frac{2 πn}{3}

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{2 πn}{3}

θ = \frac{π}{9} + \frac{2 πn}{3}

θ = \frac{π}{9} + \frac{2 πn}{3}

θ = \frac{π}{9} + \frac{2 πn}{3}

Solve

3 θ = \frac{5 π}{3} + 2 πn : θ = \frac{5 π}{9} + \frac{2 πn}{3}

3 θ = \frac{5 π}{3} + 2 πn

Divide both sides by

3

3 θ = \frac{5 π}{3} + 2 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} : \frac{5 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{5 π}{9} + \frac{2 πn}{3}

θ = \frac{5 π}{9} + \frac{2 πn}{3}

θ = \frac{5 π}{9} + \frac{2 πn}{3}

θ = \frac{5 π}{9} + \frac{2 πn}{3}

θ = \frac{π}{9} + \frac{2 πn}{3}, θ = \frac{5 π}{9} + \frac{2 πn}{3}

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{9} + \frac{2 πn}{3}, θ = \frac{5 π}{9} + \frac{2 πn}{3}

Graph

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