# Solve 2*C*sin(W)+P*cos(N*W)=P for W

• I
• SSGD
In summary, the conversation discusses an equation with multiple solutions for W, with C and P as constants and N as an integer. The speaker mentions that there are special cases where a solution is possible, but not for larger N values. They suggest converting the equation to a polynomial and using Taylor series or numerical methods to approximate a solution. The speaker also mentions that they need to solve for a large range of N values, which would require solving large polynomials.
SSGD
Is there a way to solve for W in the below equation. There has to be multiple solution for W, but I am at a loss as to how to solve this.

2*C*sin(W)+P*cos(N*W)=P or 2*C/P*sin(W)+cos(N*W)=1

C and P are constants
N is an integer

Not in a closed form for general N. There are special cases where a solution is possible, e.g. N=1.

I would expect N=2 and maybe N=3 and N=4 to have exact solutions as well, but not larger N. Careful: I didn't check those cases in detail.

You can write sine and cosine as sum of two complex exponentials, and convert your expression to a polynomial (with powers of eiW). Those polynomial equations have proper, general exact solutions only up to 4th order.

Made an error... change the sign before the P from + to -

2*C*sin(W)-P*cos(N*W)=P or 2*C/P*sin(W)-cos(N*W)=1

That doesn't matter. C and P can be arbitrary constants anyway.

Would there be a way to approximate a solution if we assume W is positive and near zero but always greater than zero.

Because I can't solve it for all the solutions. Would there be a way to find the smallest positive solution.

Taylor series might work well. You can always find solutions to arbitrary precision with numerical methods. It depends on your problem then. Do you prefer an approximation that is not so good, but can be written down as formula, or an approximation that is much better, but needs dedicated calculation in each case?

SSGD
Yeah I just did both. Taylor Series gives me a number, but I was hoping to get several decimals of accuracy. But, the number of terms I will need is going to be... I think... Large.

The numerical solution gives the right askers, but It always wants to go to the Trivial Solution "0". With the constraints it worked.

Thanks for the help on this. I didn't think there was a way to get an general analytic solution. I would need to solve it for N's in the range of 20 to 100. So we are talking about some really large polynomials.

Again that's for the help.

## 1. What is the meaning of the variables in the equation 2*C*sin(W)+P*cos(N*W)=P for W?

The variable C represents the amplitude of the sine wave, while P represents the amplitude of the cosine wave. W is the angular frequency and N is the number of cycles within the given time period.

## 2. How do you solve the equation 2*C*sin(W)+P*cos(N*W)=P for W?

First, we can rearrange the equation to isolate the sine function by subtracting P*cos(N*W) from both sides. Then, we can divide both sides by 2*C to get sin(W) alone. Finally, we can use the inverse sine function to find the value of W.

## 3. Can this equation be solved without using trigonometric functions?

No, this equation contains both sine and cosine functions, which are trigonometric functions. Therefore, trigonometry must be used to solve it.

## 4. Is there a specific method or formula to solve equations with both sine and cosine functions?

Yes, there are several methods that can be used to solve equations with both sine and cosine functions. One common method is to use the double angle formula for sine and cosine to rewrite the equation in terms of only one of the functions. Another method is to use the substitution method, where we substitute a variable for one of the trigonometric functions to create a system of equations that can be solved.

## 5. Can this equation have multiple solutions?

Yes, since the sine and cosine functions are periodic, there can be multiple solutions for W that satisfy the equation. These solutions will occur at regular intervals and can be found by adding or subtracting multiples of the period of the functions.

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