Find all angles between 0° and 180° (not inclusive) which satisfy the equation
cos x - sin 2x + cos 3x - sin 4x = 0
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Answer:
cos x - sin 2x + cos 3x - sin 4x = 0
(cos x + cos 3x) - (sin 2x + sin 4x) = 0
2 cos 2x cos x - 2 sin 3x cos x = 0
(2 cos x)(cos 2x - sin 3x) = 0
Either cos 2x = sin 3x or cos x = 0
For cos x = 0, x = 90°
For cos 2x = sin 3x,
(1 - 2 sin2 x) = 3 sin x - 4 sin3 x
4 sin3 x - 2 sin2 x - 3 sin x + 1 = 0
(4 sin2 x + 2 sin x - 1)(sin x - 1) = 0
Either
sin x = 1 (reject as 0° and 180° are not inclusive)
or
sin x = (-1 + √5) / 4 or (-1 - √5) / 4 (reject as sine is positive between 0° and 180° )
Thus, x = 18° or 162°
Combining, x = 18°, 90° or 162°
