Trigonometry – Solve tan (x) + cos (x) = 1/2 – Mathematics Stack Exchange

Is it possible (not numerical) to find that? $ x $ such as:

$$
tan (x) + cos (x) = 1/2
$$

?

All my attempts are finished in a 4-degree polynomial. Example: Calling c = cos (x):

$$
frac { sqrt {1-c ^ 2}} {c} + c = frac {1} {2}
$$

$$
sqrt {1-c ^ 2} + c ^ 2 = frac {1} {2} c
$$

$$
1-c ^ 2 = c ^ 2 ( frac {1} {2} -c) ^ 2 = c ^ 2 ( frac {1} {4} -c + c ^ 2)
$$

$$
c ^ 4-c ^ 3 + frac {5} {4} c ^ 2-1 = 0
$$

Complexity Theory – If there is no known polynomial time algorithm to solve the B-reduced A and A poly time, there is no known polynomial time algorithm to solve B

Is the statement below true or false and please give me a little explanation.

If there is no known polynomial time algorithm to solve A and A ≤ pB, there is no known polynomial time algorithm to solve B.

I think wrong, but I'm not sure.
Many Thanks!

Differential equations – How do you solve these ODEs with NDSolve?

The calculation changes t = 0 there sin[ψ

eq1 = ω1
eq2 = ω2

and construct the linear combinations,

eq1n = Simplify[eq1Sin[eq1Sin[eq1Sin[eq1Sin[ψ
(* Cos[ψ

(If this were not possible, the equations themselves could not be solved in principle.)

Now replace eq1, eq2 by eq1n, eq2n,

I1 = 2; I2 = 3; I3 = 4;
s = NDSolveValue[{I1*ω1'[{I1*ω1'[{I1*ω1'[{I1*ω1'
I2 * ω2 & # 39;
I3 * ω3 & # 39;
eq1n == 0, eq2n == 0,
ω3
ω1[0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] == 0, φ[0] == 0, θ[0] == Pi / 6},
{ω1
plot[Evaluate@s[[1 ;; 3]], {t, 0, 120}, ImageSize -> Large]plot[Evaluate@s[[4 ;; 6]], {t, 0, 120}, ImageSize -> Large]

Enter the image description here

Enter the image description here

Incidentally, the original equations can be solved by slightly changing the initial state ψ[0] == 0 to ψ[0] == 10 ^ -6,

Another approach is to use the option.

Method -> {"EquationSimplification" -> "Residual"}

Everyone gives the same answer.

How do I solve these Odes with NDSolve?

I have six odes and can not use DSolve. So I tried NDSolve. But it does mean that there may be mistakes. The code looks like this:

I1 = 2; I2 = 3; I3 = 4;
NDSolve[{I1*ω1'
I2 * ω2 & # 39;
I3 * ω3 & # 39;
0, ω1[
t] == φ & # 39;
sin[ψ[ψ[ψ[ψ
t] == φ & # 39;
cos[ψ[ψ[ψ[ψ
t] == φ & # 39;
0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] ==
0, φ[0] == 0, θ[0] ==
Pi / 6}, {ω1, ω2, ω3, ψ, φ,
θ}, {t, 0, 120}]

I want to know how to avoid this mistake.

To solve basic problems with asymptotic algorithms

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Equation Solution – Solve a nonlinear system with eight variables

Please I need urgent help
We consider the three functions
H1 = 4 (x + b1y) ^ 2 + 8 (c1x-d1y) + e1 ^ 2 y ^ 2, where e1> 0
H2 = 4 (x + b2y) ^ 2 + 8 (c2x-d2y) + e2 ^ 2 y ^ 2 with e2> 0
H3 = 4 (x + b3y) ^ 2 + 8 (c3x-d3y) + e3 ^ 2 y ^ 2 with e3> 0 … d1, b1, c1, d2, c2, b2, d3, c3, b3 are real Parameter.
I would like to solve the following system (to get the values ​​of alpha, beta, gamma, delta, f, h, g, k) or (if that is not possible) to specify the maximum number of its solutions
The system is E1 = E2 = E3 = E4 = E5 = E6 = E7 = E8 = 0, so that
E1 = H1 (alpha, beta) -H1 (gamma, delta), E2 = H2 (alpha, beta) -H2 (f, g), E3 = H2 (gamma, delta) -H2 (h, k), E4 = H3 (h, k) -H3 (f, g), E5 = beta ^ 2-alpha (alpha-1) (alpha-3), E6 = delta ^ 2-gamma (gamma-1) (gamma-3), E7 = g ^ 2-f (f-1) (f-3), E8 = k ^ 2-h (h-1) (h-3).

Solve inheritance without affecting data consistency in a relational database

In trying to make the database better, I noticed that I am always tempted to solve variations of the same problem.

Here is an example with general requirements:

  1. An online store sells various categories of product,
  2. The system must z. For example, you can get the list of all product categories foods and Furniture,
  3. A customer can order and retrieve each product assignment History.
  4. The system must store certain properties depending on the product category. say that expiration date and calories for each foods Product and manufacturing date for each Furniture Product.

Without requirement 4, the model could be quite straightforward:

Enter the image description here

Problem is the attempt to solve requirement 4. I thought of something like this:

Enter the image description here

In this approach, the product furniture and product food relationship is a supertype subtype relationship; The primary key of the subtype is also a foreign key for the supertype primary key.

However, this approach can not guarantee category referenced by a foreign key product will match the actual subtype. For example, nothing prevents me from hiring foods Category to a product tuple with a subtype record in the Furniture Table.

I've read several articles on inheritance in relational database modeling, especially these and these, which were very helpful but did not solve my problem for the reasons mentioned above. Regardless of which model I come from, I am never satisfied with the consistency of the data.

How can I solve requirement 4 without compromising data consistency? Did I go wrong here? If so, what would be the best way to solve this problem based on these requirements?

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Why does Solve block trying to solve the quardatic equation with large integers?

Why is Solve blocking when trying to solve Solve?[(x^2+y^2)+(x+y)==2^256 && x>0 && y>0,{x,y},Integers]? It works 2 ^ 185, but at higher powers of 2, processing seems to stop. The program says it works, but there is no solution if it runs overnight. Running Mathematica 11.3 on Windows 32-bit operating system.