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-5-cos-2x-0-holds-for-all-x-is-

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 \dots (1)

x = π :

a_{1} - a_{3} + a_{5} = 0 \dots (2)

(1) - (2) \Rightarrow a_{3} = 0

x = \frac{π}{2} :

a_{1} + a_{2} - a_{5} = 0 \dots (3)

x = - \frac{π}{2} :

a_{1} - a_{2} - a_{5} = 0 \dots (4)

(3) - (4) \Rightarrow a_{2} = 0

x = \frac{π}{4} :

a_{1} + a_{4} = 0 \dots (5)

x = - \frac{π}{4} :

a_{1} - a_{4} = 0 \dots (6)

(5) - (6) \Rightarrow a_{4} = 0

(5) + (6) \Rightarrow a_{1} = 0

(1) \Rightarrow a_{5} = 0

\Rightarrow (a_{1}, \dots 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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