Olympiad corner

Geometry Level 3

{ cos ⁡ x = cos ⁡ y sin ⁡ x = − sin ⁡ y sin ⁡ 1994 x + sin ⁡ 1994 y = ? \large \begin{cases} \cos x = \cos y \\ \sin x = - \sin y \\ \sin 1994x + \sin 1994y = ? \end{cases}

Details : Put the answer upto 3 decimal places


The answer is 0.000.

Akhil Bansal by Akhil Bansal , 22, India

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2 solutions

Tanishq Varshney
Nov 7, 2015

One line answer

x = 2 π − y x=2 \pi-y as sin ⁡ x \sin x is negative in fourth quadrant.

Answer is zero.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Same as y = 2 π − x y = 2\pi - x , it is found that with sample or some example of x = {..., -8, -5, -1, 0, 2, 3, 5, 7...}, it cannot satisfy all. The only way remained is x = -y or y = -x.

Lu Chee Ket - 5 years, 6 months ago
Lu Chee Ket
Dec 2, 2015

To satisfy both cos ⁡ x = cos ⁡ y \cos x = \cos y and also sin ⁡ x = − sin ⁡ y \sin x = -\sin y , it is strictly found after analysis that x = − y x = -y is the only way to be true all the ways.

Therefore sin ⁡ ( 1994 × x ) + sin ⁡ ( 1994 × − x ) \sin (1994 \times x) + \sin (1994 \times -x) = sin ⁡ ( 1994 × x ) − sin ⁡ ( 1994 × x ) \sin (1994 \times x) - \sin (1994 \times x) = 0.

Answer: 0.000 \boxed{0.000}

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