If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$,

Question

If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to

(a) $y$

(b) $y/2$

(c) $2y$

(d) $y/6$

I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be appreciated. Thanks!

Answer 1

Hint: $$\sin(x)+\sin82x)+\sin(3x)=\sin(2x)(2\cos(x)+1)=\sin(y)$$

$$\cos(x)+\cos(2x)+\cos(3x)=\cos(2x)(2\cos(x)+1)=\cos(y)$$ from here you will get

$$\tan(2x)=\tan(y)$$ con you finish?

Answer 2

Note that $$\sin(x)+\sin(2x)+\sin(3x)=\sin(2x)\,\big(1+2\cos(x)\big)$$ and $$\cos(x)+\cos(2x)+\cos(3x)=\cos(2x)\,\big(1+2\cos(x)\big)\,.$$ Thus, for the required equalities to be true, we need $$1+2\cos(x)=s\in\{-1,+1\}\text{ and }2x=\left\{\begin{array}{ll} 2k\pi+y\,,&\text{if }s=+1\\ (2k+1)\pi-y\,,&\text{if }s=-1\,. \end{array}\right.$$ for some integer $k$. This mean $$x=\left\{\begin{array}{ll} k\pi+\frac{y}{2}\,,&\text{if }s=+1\\ k\pi+\frac{\pi-y}{2}\,,&\text{if }s=-1\,. \end{array}\right.$$ The problem statement seems to suggest that (b) is the correct answer (with option $s=+1$ and $k=0$). However, there is a caveat, as Saucy O'Path mentioned.

To be precise, the possible values of $(x,y)$ are $$(x,y)=\left(\frac{\pi}{2}+m\pi,(2n+1)\pi\right)\text{ and }(x,y)=\big((2m+1)\pi,(2n+1)\pi\big)$$ where $m,n\in\mathbb{Z}$. In addition, $x=\pi$ is a solution, but if (b) is true, then $y=2x=2\pi$ would correspond to the value $x=\pi$, but this is not the case. When $x=\pi$, the only working values of $y$ are odd multiples of $\pi$.

Conclusion: This is a poorly designed problem, and should be ignored. None of the provided choices is a (solely) correct answer. Each except (c) can be correct, e.g., (a) with $(x,y)=(\pi,\pi)$, (b) with $(x,y)=\left(\frac{\pi}{2},\pi\right)$, and (d) with $(x,y)=\left(\frac{\pi}{2},3\pi\right)$.

Answer 3

Draw 3 unit vectors $e_1$, $e_2$ and $e_3$. The angle between these each of these vectors and $x$ axis is $x$, $2x$ and $3x$ correspondingly.

$x$ coordinates of this vectors would be $cos(x)$, $cos(2x)$ and $cos(3x)$.

To find the sum of these coordinates you can first add up the vectors and than the x-coordinate of the sum of these 3 vectors.

So, $cos(x) + cos(2x) + cos(3x)$ and $sin(x) + sin(2x) + sin(3x)$ are the coordinates of $e_1 + e_2 + e_3$ vector!

And it must be coordinates of unit vector $e_y$!

Now it's a geometry problem which looks much easier for me. $e_1 + e_2 + e_3$ should be pointing to the same direction as $e_2$.

$x=y=\pi$ seems to be a solution.

$x=\pi / 2, y=\pi$ is also a solution.

So (a) may hold and (b) also may hold.

Answer 4

If $z = e^{ix} = \cos x + i \sin x$

Then we have $z + z^2 + z^3 = e^{iy} = w$ (say)

Divide by $z^2$ to see that

$$z + \frac{1}{z} + 1 = \frac{w}{z^2}$$

The left side is real and thus

$$w = az^2$$

Since $|w| = |z| = 1$ we must have that $|a| = 1$

Thus $$w = \pm z^2$$

This gives rise to two equations:

$$z + z^3 = 0$$

and

$$z + 2z^2 + z^3 = 0$$

I will leave the rest to you.

And as others said, be careful that $e^{ix}$ is periodic and for the question to make sense you might need to put bounds on $x,y$.

Answer 5

Generalization :

Using How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?,

$$f(n)=\dfrac{\sum_{k=0}^{n-1}\sin (a+k \cdot d) }{\sum_{k=0}^{n-1}\cos (a+k \cdot d)}= \tan \frac{2 a + (n-1)d}2$$

If $a=d=x,$ $$f(n)=\tan\dfrac{x(n+1)}2$$

Here $n=3$