Question

solve if sinx+sin2x+sin3x=0​

Answer 1

Answer:

Use trig identity:

sin a + sin b = 2sin ((a + b)/2).cos ((a - b)/2)

In this case:

sin x + sin 3x = 2sin (2x).cos (x)

(sin x + sin 3x) + sin 2x = 2sin (2x)cos (x) + sin (2x) =

= sin (2x)(2cos x + 1) = 0

Either one of the 2 factors must be zero.

a. sin 2x = 0

2x = 0 --> x = 0

2x = pi --> x = pi/2

2x = 2pi --> x = pi

General answer: x = (kpi)/2

b. 2cos x + 1 = 0 --> cos x = - 1/2

Trig table and unit circle give 2 solutions -->

x = +- (2pi)/3 + 2kpi

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Answer 2

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$\large \tt \implies x = \frac{k\pi}{2}$

$\large \tt \implies x = + \frac{2\pi}{3 } + 2k\pi$

$\large \tt \implies sin \: x \: + \: sin \: 3x \: = 2sin(2x).cos(x)$