Find all solutions to cos ( 7 θ ) − cos ( θ ) = sin ( 4 θ ) on 0 ≤ θ < 2 π 3

Answer 1

cos 7θ - cos θ = -2 sin ((7θ + θ)/2)sin(7θ - θ)/2)

Can you simplify, set equatl to sin 4θ and answer?

Answer 2

cos ( 7 θ ) − cos ( θ ) = sin ( 4 θ ) on 0 ≤ θ < 2 π /3

-2 [sin( 7θ+θ)/2][ sin (7θ - θ)/2] = sin(4θ)

sin (4θ) + 2 [sin( 7θ+θ)/2][ sin (7θ - θ)/2] = 0

sin (4θ) + 2 sin (4θ) sin 3θ = 0

sin (4θ)[ 1 + 2 sin 3θ ]= 0 from where we deduce

sin 4θ =0 or sin3θ = - 1/2

A. From sin4θ = 0 we get 4θ = 2kπ or 4θ = 2kπ + π or equivalently θ= κπ/2 or θ= κπ/2 + π/4.

B. From sin3θ = - 1/2 we obtain 3θ= 2κπ - π/6 or 3θ= 2κπ + π+π/6 That is θ= 2κπ/3 - π/18 or

θ = 2κπ/3 + 7π/18.

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