Linear Combinations of Trig Functions - Finding Roots

In summary, the conversation is about solving for the roots of a complicated summation of trigonometric functions with different amplitudes, phases, and frequencies. The goal is to find the values of x on [0, pi/2] that satisfy the condition and the preferred solution is an analytical one rather than an approximation. The conversation explores different methods and substitutions, but concludes that there is no simple way to combine the two terms into one function.
  • #1
david_p
6
0
Hi there
I was wondering if there is a simple way to solve for the roots of a complicated summation of trig functions that can't be combined with any simple identities.
I have an equation of the form:

0 = sin(8x-arctan(4/3))+3.2sin(16x+pi/2)

where the two sines have different amplitudes, phases and a frequency that differs by a factor of 2. I would like to find the values of x on [0,pi/2] that satisfy this condition. I don't know very much about Fourier analysis but is there some method for solving this?
I need an analytical solution - not an approximation
 
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  • #2
I don't think you need to go into anything complicated. What does [tex]Sin(\theta + \frac{\pi}{2})[/tex] equal?
 
  • #3
Yes, it equals cosx but the frequency is still different so you can't easily combine the two terms. I don't think there is any way of combining the two terms into one simple function. The solution must have an amplitude that oscillates as a function of x; like a form:
Asin(Bx+C)sin(Dx+E)

Not sure how to solve for the parameters though
 
  • #4
Oh right, I didn't see that, maybe this isn't as simple as it looks. I'm assuming you've exausted double angle and angle addition formulas?
 
  • #5
Yes, I've tried almost every standard substitution. I don't think this will solve the problem since the amplitude varies with x. Plotting the function in maple reveals that it doesn't have the form of a simple cosine function
 
  • #6
you could still use the cosx if you treat it as sin(16x+pi/2) = cos(16x) = cos(2*8x) = 2cos^2(8x)-1
whereas you could expand the first term sin(8x- ... getting it in terms of sin8x and cos8x, substituting sin8x=sqrt(1-cos^2(8x)) and squaring to eliminate the sqrt leaves you with a 4th degree polynomial of cos8x. which is technically solvable

if i thought it through enough, didn't try in written
 
  • #7
No this won't work either, you will end up with a function that has non integer cos terms
Unless I misread what you were saying
 
  • #8
david_p said:
I need an analytical solution - not an approximation
Why? What is your application?
 
  • #9
Its an assignment question,
but I'm also curious if there is a generalized technique for approaching this type of problem
 
  • #10
david_p said:
No this won't work either, you will end up with a function that has non integer cos terms
Unless I misread what you were saying

you don't need integer terms to solve a 4th degree polynomial, though you can just multiply the eq with sth to make them integer

[tex]
Sin(8x-arctan(4/3))+3.2Sin(16x+\pi/2)= \pm 0.6 \sqrt{1-Cos^2(8x)}-0.8Cos(8x)+6.4Cos^2(8x)-3.2 = 0
[/tex]

or
[tex] \pm 3 \sqrt{1-a^2}-4 a+32a^2-16=0 [/tex]
where a=cos(8x)
 
Last edited:

1. What are linear combinations of trig functions?

Linear combinations of trig functions are expressions that involve adding or subtracting multiples of trigonometric functions, such as sine, cosine, tangent, etc. These combinations can help simplify and solve complex trigonometric equations.

2. How do you find the roots of a linear combination of trig functions?

To find the roots of a linear combination of trig functions, you must set the expression equal to zero and then solve for the variable. This can be done by using algebraic techniques, such as factoring or the quadratic formula, and then applying trigonometric identities to simplify the equation.

3. Can you use the unit circle to find the roots of a linear combination of trig functions?

Yes, the unit circle can be used to find the roots of a linear combination of trig functions. By substituting the values of the unit circle, which represent the coordinates of points on the circle, into the equation, you can solve for the roots.

4. Are there any special cases when finding the roots of linear combinations of trig functions?

Yes, there are a few special cases when finding the roots of linear combinations of trig functions. These include when the coefficients of the trigonometric functions are equal, when the coefficients are in a specific ratio, and when the trigonometric functions are raised to a power greater than one.

5. How can linear combinations of trig functions be applied in real-life situations?

Linear combinations of trig functions can be used to model and solve various real-life problems, such as calculating the height of a building or a mountain using trigonometric ratios, determining the position of objects in motion, and analyzing the behavior of waves and sound. They are also commonly used in fields such as engineering, physics, and astronomy.

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