Finding the solution for trigonometric equations

Question

If I have a such equation to solve: $$\sin(3x) = 1/2.$$ How can I solve this equation without using a calculator?

The app Symbolab calculated the result by using the "General Solution": $$3x=\frac{\pi}{6}+2n\pi$$ and $$3x=\frac{5\pi}{6}+2n\pi$$, however I am not sure how to obtain such General Solution? Is there a formula for this?

Thanks!

Answer 1

Within the range $[0,2\pi] \ , \sin\frac\pi6 = \sin \frac{5\pi}6 = \frac12$

So, $\sin(3x) = \sin\frac{\pi}{6} = \sin\frac{5\pi}6$

Also $\text{sine}$ has periodicity of $2\pi$. Thus $\sin(2n\pi+\theta ) =\sin\theta$ where $n$ is an integer.

So, $$\sin(3x) = \sin\frac\pi6 = \sin\left(2n\pi+\frac\pi6\right) \implies 3x = 2n\pi+\frac\pi6$$

Similarly, $$\sin(3x) = \sin\frac{5\pi}6 = \sin\left(2n\pi+\frac{5\pi}6\right) \implies 3x = 2n\pi+\frac{5\pi}6$$

Answer 2

When $$\sin x= \sin \alpha \implies x=n\pi+(-1)^n \alpha,$$ where $n=0,\pm 1,\pm 2,\pm 3,...$ So here $$\sin 3x =\sin \pi/6 \implies 3x=n\pi+(-1)^n \frac{\pi}{3} \implies x=\frac{n\pi}{3}+(-1)^n \frac{\pi}{9}, n=0, \pm 1, \pm 3,...$$

Answer 3

$sin(x)$ is periodic, as in it has a period of $2\pi$. Therefore, it will intersect the same y-value every period, or every $2n\pi$ where $n$ is an integer. By scaling $\theta$ by 3, you are actually scaling the period of $sin\theta$ by $\frac{1}{3}$. Thus the new period is $\frac{2\pi}{3}$. You can see this in action with your solution you gave, which was $$sin\theta = \frac{1}{2}$$ $$\theta = sin^{-1}(\frac{1}{2})$$ Here's the kicker: $sin\theta$ intersects $\frac{1}{2}$ twice every period, so there are two solutions. By inspecting the unit circle, find why this must be true. The two solutions are $$3\theta = \frac{\pi}{6}$$ $$3\theta = \frac{5\pi}{6}$$ But we want to know the solutions for the entire domain $(-\infty, \infty)$, so we want $$3\theta = \frac{\pi}{6} + 2n\pi$$ $$3\theta = \frac{5\pi}{6}+ 2n\pi$$ And now by dividing by $3$, you can see why your period is scaled by $\frac{1}{3}$ (as well as the solution): $$\theta = \frac{\pi}{18} + \frac{2n\pi}{3}$$ $$\theta = \frac{5\pi}{18}+ \frac{2n\pi}{3}$$