What is x  if sin8x=sin5x ?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll recall the method of solving the elementary trigonometric equation:

sin x = a => x = (-1)^k*arcsin a + k*pi

Comparing, we'll get:

8x = (-1)^k*arcsin (sin 5x) + k*pi

But arcsin (sin 5x) = 5x

8x = (-1)^k* 5x + k*pi

We'll discuss two cases:

1) If k is an even integer number:

k = 2n

8x = (-1)^2n* 5x + 2n*pi

8x = 5x + 2n*pi

8x - 5x = 2n*pi

3x = 2n*pi

x = 2n*pi/3

2) If k is an odd integer number:

k = 2n + 1

8x = (-1)^(2n+1)* 5x + (2n+1)*pi

8x = -5x + 2n*pi + pi

8x + 5x = 2n*pi + pi

13x = 2n*pi + pi

x = 2n*pi/13 + pi/13

The solutions of the equation are represented by the reunion of the sets: {2n*pi/3/ n is an integer number} U {(2n+1)*pi/13 / n is an integer number}.

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