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Advanced Math/position of the nodes on a waveform


I have a waveform of y1(x)= 1.2*cos(1.2*x)+1.8*cos(1.8*x), and i need to find the position of the nodes.

I found the nodes for y2(x)= sin(x*1.2)+sin(x*1.8) with either x = nπ/(1.5) or x = (2n+1)*(π/0.6), where n = 0, 1,2 3, and so on.

but i can't find an easy way for the first one.

If we let t = 0.6x, then x = t /0.6 and we get y1(t/0.6) = 1.2*cos(2t) + 1.8*cos(3t).

Knowing that cos(a+b) = cos(a)cos(b) - sin(a)sin(b) with a=t and b=2t, we get
so cos(3t) = cos(2t)cos(t) - sin(2t)sin(t), which giveds
y1(t/0.6) = 1.2*cos(2t) + 1.8*cos(t)*cos(2t) - 1.8*sin(t)sin(2t).

Factoring out the cos(2t) from the first two terms gives
y1(5t/3) = (1.2 + 1.8*cos(t))cos(2t) - 1.8*sin(t)sin(2t).

It is also known that cos(2t) = 2*cos²(t) - 1 and sin(2t) = 2*sin(t)cos(t).
This gives y1(5t/3) = (1.2 + 1.8*cos(t))(2*cos²(t) - 1) - 1.8*sin(t)*2*sin(t)cos(t).

The 1st term of y1 is (1.2 + 1.8*cos(t))(2*cos²(t) - 1), and this is
3.6*cos³(t) + 2.4*cos²(t) - 1.8*cos(t) - 1.2.

That 2nd term is 1.8*sin(t)*2*sin(t)cos(t), and this is
3.6*sin²(t)cos(t) = 3.6*(1-cos²(t))cos(t) = 3.6*cos(t)-3.6*cos³(t)).

Combining the 1st and 2nd terms cancels the 3.6*cos³(t), giving
2.4*cos²(t) + 1.8*cos(t) - 1.2.

Letting u = cos(t), this is a quadratic in u, which is 2.4u² + 1.8u - 1.2.
Rewriting this gives (3/5)(4u² + 3u - 2).

The solution to 4u² + 3u - 2 is given by u = (-3±5)/8, which gives u=1/4 or u=-1.

Knowing that u=cos(t), this can be used to find t.
Note that if u=-1, t=(2n+1)π for any integer n and also
if u=1/4, t = arccos(1/4), which gives another set of numbers that are 2π apart.

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Scott A Wilson


I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?


Experience in the area; I have tutored people in the above areas of mathematics for over two years in I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.

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