Need help with solving Trigonometric Equation

[student95]

Good evening,

I've been asked to find the solutions to 2sin(4x) = ±0.5, 0<x<π
The solutions must be in radians and I am to sketch a graph of the function with all relevant points. I have some idea with regards to solving it and I've attached a pic of my progress. I just need help finding the other solutions and drawing the sine graph.

Thank you kindly and much appreciated for your time.

 

[MarkFL]

Hello, and welcome to FMH!

I would write:

[MATH]\sin(4x)=\pm\frac{1}{4}[/MATH]
Can you see this implies:

[MATH]4x=\pi k\pm\arcsin\left(\frac{1}{4}\right)[/MATH] where \(k\in\mathbb{Z}\) ?

 

[student95]

So I would use this to find the values for x?

 

[Steven G]

Do you understand why 4x=πk±arcsin(1/4) where k∈Z ?

If you know what 4x equals then how do you find out what x equals.

 

[MarkFL]

So I would use this to find the values for x?

Yes, what I posted implies:

[MATH]x=\frac{1}{4}\left(\pi k\pm\arcsin\left(\frac{1}{4}\right)\right)[/MATH]
We are told also:

[MATH]0<x<\pi[/MATH]
[MATH]0<\frac{1}{4}\left(\pi k\pm\arcsin\left(\frac{1}{4}\right)\right)<\pi[/MATH]
[MATH]0<\pi k\pm\arcsin\left(\frac{1}{4}\right)<4\pi[/MATH]
[MATH]0<k\pm\frac{1}{\pi}\arcsin\left(\frac{1}{4}\right)<4[/MATH]
[MATH]\frac{1}{\pi}\arcsin\left(\frac{1}{4}\right)<k<4-\frac{1}{\pi}\arcsin\left(\frac{1}{4}\right)[/MATH]
This implies (given that \(k\) is an integer):

[MATH]k\in\{1,2,3\}[/MATH]
This gives you 6 solutions.

 

[student95]

Do you understand why 4x=πk±arcsin(1/4) where k∈Z ?

If you know what 4x equals then how do you find out what x equals.

I divide by 4x by 4 to find x. Apologies, I haven't attempted math since my childhood and I'm trying to grasp the concepts for a module in higher education.

 

[HallsofIvy]

You don't just "divide 4x by 4", you divide both sides of the equation by 4. The crucial thing is what you get on the other side of the equation!