If I have a such equation to solve:\sin 3x=\frac12

Question

If I have a such equation to solve:\sin 3x=\frac12

guringpw 2021-12-26 Answered

If I have a such equation to solve:
$\sin 3 x = \frac{1}{2}$

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Expert Answer

Philip Williams

Answered 2021-12-27 Author has 39 answers

Within the range

[0, 2 π], \sin \frac{π}{6} = \sin \frac{5 π}{6} = \frac{12}{}

So,

\sin (3 x) = \sin \frac{π}{6} = \sin \frac{5 π}{6}

Also sine has periodicity of

2 π

. Thus

\sin (2 n π + θ) = \sin θ

where n is an integer.
So,

\sin (3 x) = \sin \frac{π}{6} = \sin (2 n π + \frac{π}{6}) \Rightarrow 3 x = 2 n π + \frac{π}{6}

Similarly,

\sin (3 x) = \sin \frac{5 π}{6} = \sin (2 n π + \frac{5 π}{6}) \Rightarrow 3 x = 2 n π + \frac{5 π}{6}

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Jim Hunt

Answered 2021-12-28 Author has 45 answers

When

\sin x = \sin α \Rightarrow x = n π + {(- 1)}^{n} α

,
where

n = 0, \pm \pm 1, \pm 2, \pm 3, \dots

So here

\sin 3 x = \sin \frac{π}{6} \Rightarrow 3 x = n π + {(- 1)}^{n} \frac{π}{3} \Rightarrow x = \frac{n π}{3} + {(- 1)}^{n} \frac{π}{9}, n = 0, \pm 1, \pm 3, \dots

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nick1337

Answered 2022-01-08 Author has 510 answers

$\sin (x)$ is periodic, as in it has a period of $2 π$ . Therefore, it will intersect the same y-value every period, or every $2 n π$ where n is an integer. By scaling $θ$ by 3, you are actually scaling the period of $\sin θ$ by 13. Thus the new period is $\frac{2 π}{3}$ . You can see this in action with your solution you gave, which was
$\sin θ = \frac{1}{2}$
$θ = \sin^{- 1} (\frac{1}{2})$
Here's the kicker: $\sin θ$ intersects 12 twice every period, so there are two solutions. By inspecting the unit circle, find why this must be true. The two solutions are
$3 θ = \frac{π}{6}$
$3 θ = \frac{5 π}{6}$
But we want to know the solutions for the entire domain $(- \infty, \infty)$ , so we want
$3 θ = \frac{π}{6} + 2 n π$
$3 θ = \frac{5 π}{6} + 2 n π$
And now by dividing by 3, you can see why your period is scaled by $\frac{1}{3}$ (as well as the solution):
$θ = \frac{π}{18} + \frac{2 n π}{3}$
$θ = \frac{5 π}{18} + \frac{2 n π}{3}$

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