What is the general solution for cos(5θ)=cos(3θ) ?

The general solution for cos(5θ)=cos(3θ) is θ=(pin)/2 ,θ= pi/4+(pin)/2 ,θ=2pin,θ=pi+2pin

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cos(5θ)=cos(3θ)

Solution

cos (5 θ) = cos (3 θ)

Solution

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

Degrees

θ = 0^{\circ} + 9 0^{\circ} n, θ = 4 5^{\circ} + 9 0^{\circ} n, θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (5 θ) = cos (3 θ)

Subtract

cos (3 θ)

from both sides

cos (5 θ) - cos (3 θ) = 0

Rewrite using trig identities

- cos (3 θ) + cos (5 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{5 θ + 3 θ}{2}) sin (\frac{5 θ - 3 θ}{2})

Simplify

- 2 sin (\frac{5 θ + 3 θ}{2}) sin (\frac{5 θ - 3 θ}{2}) : - 2 sin (4 θ) sin (θ)

- 2 sin (\frac{5 θ + 3 θ}{2}) sin (\frac{5 θ - 3 θ}{2})

\frac{5 θ + 3 θ}{2} = 4 θ

\frac{5 θ + 3 θ}{2}

Add similar elements:

5 θ + 3 θ = 8 θ

= \frac{8 θ}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 θ

= - 2 sin (4 θ) sin (\frac{5 θ - 3 θ}{2})

\frac{5 θ - 3 θ}{2} = θ

\frac{5 θ - 3 θ}{2}

Add similar elements:

5 θ - 3 θ = 2 θ

= \frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

= - 2 sin (4 θ) sin (θ)

= - 2 sin (4 θ) sin (θ)

- 2 sin (4 θ) sin (θ) = 0

Solving each part separately

sin (4 θ) = 0 or sin (θ) = 0

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

sin (4 θ) = 0

General solutions for

sin (4 θ) = 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}

4 θ = 0 + 2 πn, 4 θ = π + 2 πn

4 θ = 0 + 2 πn, 4 θ = π + 2 πn

Solve

4 θ = 0 + 2 πn : θ = \frac{πn}{2}

4 θ = 0 + 2 πn

0 + 2 πn = 2 πn

4 θ = 2 πn

Divide both sides by

4

4 θ = 2 πn

Divide both sides by

4

\frac{4 θ}{4} = \frac{2 πn}{4}

Simplify

θ = \frac{πn}{2}

θ = \frac{πn}{2}

Solve

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

4 θ = π + 2 πn

Divide both sides by

4

4 θ = π + 2 πn

Divide both sides by

4

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

Simplify

θ = \frac{π}{4} + \frac{πn}{2}

θ = \frac{π}{4} + \frac{πn}{2}

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

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(5θ)=cos(3θ) ?
The general solution for cos(5θ)=cos(3θ) is θ=(pin)/2 ,θ= pi/4+(pin)/2 ,θ=2pin,θ=pi+2pin