how can solve δ2?

 

Graphical solution -

http://www.wolframalpha.com/input/?i=cosx++.7x+=+1.186

I looked for identities to simplify, failed to find a useful one.


Regards, Dana.

 

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how can solve δ2?

There isn't going to be a closed form solution. IF it turns out that δ2 is sufficiently small, then you can get an approximate solution by noting that, for small θ,

cos(θ) ≈ 1 - θ²

From there, you can either use a graphical approach or a numerical approach.

The graphical approach would be to note that you can set this up as two equations in δ2 that have to be equal

cos(δ2) = 1.186 - 0.7·δ2

Plot both equations and see where they intersect. Note that it is possible that they intersect in more than one place.

For the numerical approach, you can use something like Newton-Raphson, binary search, a spreadsheet's goal seek functionality, or just guided guessing.

 

thanks for the help,

how to change the cos(pi-δ) to -cos(δ)

 

There isn't going to be a closed form solution.

Curiosity: how do you know this?

(It's been years since I studied transcendental math & identities beyond those used in my day-to-day work.)

 

Curiosity: how do you know this?

(It's been years since I studied transcendental math & identities beyond those used in my day-to-day work.)

I can't give you a hard answer. In part that's because "closed form" doesn't have a simple definition since it has meaning only within the context of a given set of functions and operators. Hence one could always define a function that gives the solution to a given problem and then claim that the problem has a closed form solution.

While that might sound facetious, it's really not. Most, if not all, of the transcendental functions we use all the time were defined specifically to allow us to express the solutions to common problems in closed form. That goes for the trig functions, the log functions, the hyperbolic functions, the Bessel functions, the (pretty much you name it) functions.

But, in the vast majority of cases, when we mix transcendental and algebraic terms within an expression, the resulting solutions cannot be written in closed form over the set of the commonly recognized functions and operators. There can certainly be exceptions, but they almost always have to be very carefully crafted and the given expression shows no indication of falling in that category. I could be wrong.

 

thanks for the help,

how to change the cos(pi-δ) to -cos(δ)

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I have no idea what the δcr is in your list of expressions.

But draw the unit circle, draw a relatively small angle δ (say something ~30°) and then draw the angle (π-δ) and then determine the cosine of both. How do they relate?

It should also be a trivial matter to use the classic trig identity for the cosine of the sum/difference of two angles, along with the knowledge of what sin(π) and cos(π) are.