How do you solve Sin x + Sin 2x + Sin 3x + Sin 4x = 0?

May 10, 2015

In this way, using the sum-to-product formula for sinus and cosine:

$\sin \alpha + \sin \beta = 2 \sin \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right)$

$\cos \alpha + \cos \beta = 2 \cos \left(\frac{\alpha + \beta}{2}\right) \cos \left(\frac{\alpha - \beta}{2}\right)$.

So:

$\left(\sin 4 x + \sin x\right) + \left(\sin 3 x + \sin 2 x\right) = 0 \Rightarrow$

$2 \sin \left(\frac{4 x + x}{2}\right) \cos \left(\frac{4 x - x}{2}\right) +$

$+ 2 \sin \left(\frac{3 x + 2 x}{2}\right) \cos \left(\frac{3 x - 2 x}{2}\right) = 0$

$2 \sin \left(\frac{5}{2} x\right) \cos \left(\frac{3}{2} x\right) + 2 \sin \left(\frac{5}{2} x\right) \cos \left(\frac{1}{2} x\right) = 0$

$2 \sin \left(\frac{5}{2} x\right) \left[\cos \left(\frac{3}{2} x\right) + \cos \left(\frac{1}{2} x\right)\right] = 0$

$2 \sin \left(\frac{5}{2} x\right) 2 \cos \left(\frac{\frac{3}{2} x + \frac{1}{2} x}{2}\right) \cos \left(\frac{\frac{3}{2} x - \frac{1}{2} x}{2}\right) = 0$

$4 \sin \left(\frac{5}{2} x\right) \cos x \cos \left(\frac{x}{2}\right) = 0$.

Then:

$\sin \left(\frac{5}{2} x\right) = 0 \Rightarrow \frac{5}{2} x = k \pi \Rightarrow x = \frac{2}{5} k \pi$,

$\cos x = 0 \Rightarrow x = \frac{\pi}{2} + k \pi$,

$\cos \left(\frac{x}{2}\right) = 0 \Rightarrow \frac{x}{2} = \frac{\pi}{2} + k \pi \Rightarrow x = \pi + 2 k \pi$.

May 23, 2015

Reminder of Concept to solve trig equations:
To solve a complex trig equation, transform it into a few basic trig equations. Solving trig equations finally results in solving basic trig equations.
In this example of solving: f(x) = sin x + sin 2x + sin 3x + sin 4x = 0, use the trig identity of "sin a + sin b" to transform f(x) into a product of 3 basic trig equations like Massimiliano finely did:
$f \left(x\right) = 4 \cos x . \sin \left(5 \frac{x}{2}\right) . \cos \left(\frac{x}{2}\right) = 0$.
Next, solve separately the 3 basic trig equations: cos x = 0, sin $\left(5 \frac{x}{2}\right) = 0 , \mathmr{and} \cos \left(\frac{x}{2}\right) = 0$