The number of all possible 5−tuples (a_1 , a_2 , a_3 , a_4 , a_5 ) such that a_1 +a_2 sin x+a_3 cos x+a_4 sin 2x+a


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Question Number 44128 by gunawan last updated on 22/Sep/18

$The number of all possible 5 - tuples$ $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) such that$ $a_{1} + a_{2} \sin x + a_{3} \cos x + a_{4} \sin 2 x + a_{5} \cos 2 x = 0$ $holds for all x is$
	Answered by MrW3 last updated on 22/Sep/18

	$W a y 1 t o s o l v e :$ $x = 0 :$ $a_{1} + a_{3} + a_{5} = 0 . . . (1)$ $x = π :$ $a_{1} - a_{3} + a_{5} = 0 . . . (2)$ $(1) - (2) \Rightarrow a_{3} = 0$ $x = \frac{π}{2} :$ $a_{1} + a_{2} - a_{5} = 0 . . . (3)$ $x = - \frac{π}{2} :$ $a_{1} - a_{2} - a_{5} = 0 . . . (4)$ $(3) - (4) \Rightarrow a_{2} = 0$ $x = \frac{π}{4} :$ $a_{1} + a_{4} = 0 . . . (5)$ $x = - \frac{π}{4} :$ $a_{1} - a_{4} = 0 . . . (6)$ $(5) - (6) \Rightarrow a_{4} = 0$ $(5) + (6) \Rightarrow a_{1} = 0$ $(1) \Rightarrow a_{5} = 0$ $\Rightarrow (a_{1}, . . . a_{5}) = (0, 0, 0, 0, 0)$ $W a y 2 t o s o l v e :$ $a_{1} + \sqrt{a_{2}^{2} + a_{3}^{2}} \sin (x + θ_{1}) + \sqrt{a_{4}^{2} + a_{5}^{2}} \sin (2 x + θ_{2}) = 0$ $\sqrt{a_{2}^{2} + a_{3}^{2}} = 0 \Rightarrow a_{2} = 0, a_{3} = 0$ $\sqrt{a_{4}^{2} + a_{5}^{2}} = 0 \Rightarrow a_{4} = 0, a_{5} = 0$ $\Rightarrow a_{1} = 0$
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