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Answers by Richard P.

sin(x/2) + cos(x/2) = ? (answer)

You need the half angle formulas.   sin(x/2) = sqrt( (1-cos(x) ) /2 )   cos(x/2) = sqrt( (1 + cos(x) )/2)   There can be a ± in front of the square roots in some situations.  

Maximum value of (answer)

This problem can be solved very easily with calculus.     At the maximum value of θ ,  the derivative of the expression will be zero.    This derivative is    12 cos(θ) - 18 sin(θ) cos(θ)   setting this to zero results in 12...

explain remainder theorem (answer)

The polynomial remainder theorem states that if a polynomial , f(x),  is divided by (x -a) using long division of polynomials, the remainder will be f(a).    For example  if the polynomial is   f(x) = x2 -9      and we divide by x-2,  ...

help with voltage and electricity c/o physical science (answer)

The first statement is essentially true.   The conduction band electrons in single domain of iron do have parallel spins.  Since the electron has a magnetic moment, a magnetic field results.   The second is almost true. A voltage will be induced in a wire coil moving...

PDE equation (answer)

A special solution can be fairly easily guessed.   It is   f =  (1/(w -k)) cos(kx -wt) + C.   To this we would have to add a general solution to the homogenous equation  df/dx + df/dt = 0   This general solution looks like  ...

Area between curves (answer)

Rather than track your steps, I will show you how I would approach this problem.  Areas can be computed as integrals over either x or y.    For the problem at hand, it is easier to think of x as function of y and to carry out the integral over y.  Notice that the equations...

Evaluate the integral? (answer)

This integral is the evaluation of the integrand e^x^3  on the area in the x,y plane bounded by the x axis, the line  x = 1 and the curve   y = x^2    .   Integrals like this can written as either   ∫ dy ∫ dx   or    ∫...

Integrate f(x, y, z)? (answer)

There are two ways do evaluate this volume.   The first is to use the high school formula for the volume of a pyramid.  This is V = (1/3) base_area x height. The base area (in the x y plane) is 1/2.  The height = 3,   so   V =  (1/3) (1/2) 3 ...

How to write an iterated integral? (answer)

For a)    the integral is  ∫ dy ∫ dx  1     with the following limits:                  The lower limit of the outer integral is 0                  ...