Trig Model

natalie2155

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May 7, 2019
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I am trying to solve part 1 and 2 of the wave pool investigation.
The answers are 1. 0.5596 and 2. 8 secs
When I graph y= h1 +h2 I can see that the solution is correct graphically. But I'm not sure how to find it algebricly. I've tried using the compound angle formula to expand h2 then simplify the expression for h1+h2 but can't seem to make it work.

Part 2 I don't know how to start.

Thanks for your help!

15572862878565081656629960957012.jpg
 
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Hello, and welcome to FMH! :)

I would first consider:

[MATH]\sin\left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\sin(x)-\frac{1}{\sqrt{2}}\cos(x)[/MATH]
When adding the two sinusoidal functions, you should get a composite function of the form:

[MATH]h(t)=A\cos\left(\frac{\pi}{4}t\right)+B\sin\left(\frac{\pi}{4}t\right)[/MATH]
Can you find the exact function?
 
Hello, and welcome to FMH! :)

I would first consider:

[MATH]\sin\left(x-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}\sin(x)-\frac{1}{\sqrt{2}}\cos(x)[/MATH]
When adding the two sinusoidal functions, you should get a composite function of the form:

[MATH]h(t)=A\cos\left(\frac{\pi}{4}t\right)+B\sin\left(\frac{\pi}{4}t\right)[/MATH]
Can you find the exact function?

15572894964475124139840994395499.jpg
 
Yes, that's correct...let's simplify a bit and factor to state:

[MATH]h(t)=\frac{1}{5}\left((\sqrt{2}+1)\sin\left(\frac{\pi}{4}t\right)-\sqrt{2}\cos\sin\left(\frac{\pi}{4}t\right)\right)[/MATH]
Now, let's consider a sinusoid of the form:

[MATH]f(x)=A\sin(x)-B\cos(x)[/MATH]
Now suppose we let:

[MATH]A=C\cos(u)[/MATH]
[MATH]B=C\sin(u)[/MATH]
And so we have:

[MATH]f(x)=C(\sin(x)\cos(u)-\cos(x)\sin(u))[/MATH]
Can you use the angle difference identity for sine to write that strictly as a sine function?
 
Okay, excellent so far. Now consider the definitions for \(A\) and \(B\)...what if you square them and add them together. What if you divide the latter by the former...what do you conclude from these two separate operations?
 
Not too sure about the next step.
 

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Actually I figured out part 1. :)
What about question 2?
 

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Actually I have worked out the second part too. Thank you very much for your help MarkFL
 

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