Question

Sin x + sin 3x + cos x = 0​

Answer 1

$\sin(x)+\sin(3x)+\cos(x)=0\\2\sin(2x)\cos(x)+\cos(x)=0\\\cos(x)(2\sin(2x)+1)=0\\\begin{cases}\cos(x)=0\\\sin(2x)=-\dfrac{1}{2}\end{cases}\Rightarrow\begin{cases}x=\dfrac{\pi}{2}+\pi k\\2x=-\dfrac{\pi}{6}+2\pi k\\2x=-\frac{7\pi}{6}+2\pi k\end{cases}, k \in \mathbb Z\Rightarrow\begin{cases}x=\dfrac{\pi}{2}+\pi k\\x=-\dfrac{\pi}{12}+\pi k\\x=-\frac{7\pi}{12}+\pi k\end{cases}k\in\mathbb Z$

Answer 2

To solve the equation sin x + sin 3x + cos x = 0, we can use trigonometric identities and algebraic techniques. First, let's rewrite the equation using trigonometric identities:

sin 3x = 3 sin x - 4 sin^3 x

Now the equation becomes:

sin x + (3 sin x - 4 sin^3 x) + cos x = 0

Combine like terms:

4 sin^3 x - 2 sin x + cos x = 0

Next, we can use the identity cos x = 1 - sin^2 x to express the equation solely in terms of sin x:

4 sin^3 x - 2 sin x + (1 - sin^2 x) = 0

Now, we have a cubic equation in sin x:

4 sin^3 x - 2 sin x + 1 - sin^2 x = 0

Rearrange the terms:

4 sin^3 x - sin^2 x - 2 sin x + 1 = 0

This is a cubic equation, and solving it directly for sin x can be challenging. However, we can try to find some simple solutions.

One solution can be sin x = 0:

sin x = 0

This gives us x = 0° and x = 180° as possible solutions.

Next, let's look for possible integer solutions of sin x (sin x = ±1/2) since it is easier to work with:

1. When sin x = 1/2: 2 sin x = 1 sin x = 1/2

This gives us x = 30° and x = 150° as possible solutions.

2. When sin x = -1/2: 2 sin x = -1 sin x = -1/2

This gives us x = 210° and x = 330° as possible solutions.

Now, we have a few possible solutions: x = 0°, x = 30°, x = 150°, x = 180°, x = 210°, and x = 330°.

However, these are just some of the possible solutions. The cubic equation might have other solutions that are not apparent using these simple approaches. To find the complete solution set, numerical methods or advanced algebraic techniques would be required.