What is the general solution for cos(4θ)=cos(3θ) ?

The general solution for cos(4θ)=cos(3θ) is θ=(4pin)/7 ,θ=(2pi)/7+(4pin)/7 ,θ=2pi+4pin

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Popular Trigonometry >

cos(4θ)=cos(3θ)

Solution

cos (4 θ) = cos (3 θ)

Solution

θ = \frac{4 πn}{7}, θ = \frac{2 π}{7} + \frac{4 πn}{7}, θ = 2 π + 4 πn

Degrees

θ = 0^{\circ} + 102.85714 \dots^{\circ} n, θ = 51.42857 \dots^{\circ} + 102.85714 \dots^{\circ} n, θ = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (4 θ) = cos (3 θ)

Subtract

cos (3 θ)

from both sides

cos (4 θ) - cos (3 θ) = 0

Rewrite using trig identities

- cos (3 θ) + cos (4 θ)

Use the Sum to Product identity:

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

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

Simplify

- 2 sin (\frac{4 θ + 3 θ}{2}) sin (\frac{4 θ - 3 θ}{2}) : - 2 sin (\frac{7 θ}{2}) sin (\frac{θ}{2})

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

Add similar elements:

4 θ + 3 θ = 7 θ

= - 2 sin (\frac{7 θ}{2}) sin (\frac{4 θ - 3 θ}{2})

Add similar elements:

4 θ - 3 θ = θ

= - 2 sin (\frac{7 θ}{2}) sin (\frac{θ}{2})

= - 2 sin (\frac{7 θ}{2}) sin (\frac{θ}{2})

- 2 sin (\frac{7 θ}{2}) sin (\frac{θ}{2}) = 0

Solving each part separately

sin (\frac{7 θ}{2}) = 0 or sin (\frac{θ}{2}) = 0

sin (\frac{7 θ}{2}) = 0 : θ = \frac{4 πn}{7}, θ = \frac{2 π}{7} + \frac{4 πn}{7}

sin (\frac{7 θ}{2}) = 0

General solutions for

sin (\frac{7 θ}{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{7 θ}{2} = 0 + 2 πn, \frac{7 θ}{2} = π + 2 πn

\frac{7 θ}{2} = 0 + 2 πn, \frac{7 θ}{2} = π + 2 πn

Solve

\frac{7 θ}{2} = 0 + 2 πn : θ = \frac{4 πn}{7}

\frac{7 θ}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{7 θ}{2} = 2 πn

Multiply both sides by

2

\frac{7 θ}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 θ}{2} = 2 \cdot 2 πn

Simplify

7 θ = 4 πn

7 θ = 4 πn

Divide both sides by

7

7 θ = 4 πn

Divide both sides by

7

\frac{7 θ}{7} = \frac{4 πn}{7}

Simplify

θ = \frac{4 πn}{7}

θ = \frac{4 πn}{7}

Solve

\frac{7 θ}{2} = π + 2 πn : θ = \frac{2 π}{7} + \frac{4 πn}{7}

\frac{7 θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{7 θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 θ}{2} = 2 π + 2 \cdot 2 πn

Simplify

7 θ = 2 π + 4 πn

7 θ = 2 π + 4 πn

Divide both sides by

7

7 θ = 2 π + 4 πn

Divide both sides by

7

\frac{7 θ}{7} = \frac{2 π}{7} + \frac{4 πn}{7}

Simplify

θ = \frac{2 π}{7} + \frac{4 πn}{7}

θ = \frac{2 π}{7} + \frac{4 πn}{7}

θ = \frac{4 πn}{7}, θ = \frac{2 π}{7} + \frac{4 πn}{7}

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

sin (\frac{θ}{2}) = 0

General solutions for

sin (\frac{θ}{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{θ}{2} = 0 + 2 πn, \frac{θ}{2} = π + 2 πn

\frac{θ}{2} = 0 + 2 πn, \frac{θ}{2} = π + 2 πn

Solve

\frac{θ}{2} = 0 + 2 πn : θ = 4 πn

\frac{θ}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{θ}{2} = 2 πn

Multiply both sides by

2

\frac{θ}{2} = 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn

Simplify

θ = 4 πn

θ = 4 πn

Solve

\frac{θ}{2} = π + 2 πn : θ = 2 π + 4 πn

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 π + 2 \cdot 2 πn

Simplify

θ = 2 π + 4 πn

θ = 2 π + 4 πn

θ = 4 πn, θ = 2 π + 4 πn

Combine all the solutions

θ = \frac{4 πn}{7}, θ = \frac{2 π}{7} + \frac{4 πn}{7}, θ = 4 πn, θ = 2 π + 4 πn

Merge Overlapping Intervals

θ = \frac{4 πn}{7}, θ = \frac{2 π}{7} + \frac{4 πn}{7}, θ = 2 π + 4 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(4θ)=cos(3θ) ?
The general solution for cos(4θ)=cos(3θ) is θ=(4pin)/7 ,θ=(2pi)/7+(4pin)/7 ,θ=2pi+4pin