Find all the values of x in the

2022-03-28

Find all the values of x in the interval [0,2π] that satisfies the inequality -1

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asked 2022-03-24
Let x and y be real numbers such that sinxsiny=3 and cosxcosy=12. The value of sin2xsin2y+cos2xcos2y can be expressed in the form pq, where p and q are relatively ' positive integers. Find p+q.
asked 2022-05-21
M = 1 + a b
S = a + b
So, I put it up like this:
1 + a b = a + b
If M = S, how to isolate a?

New questions

Find all triangles with a fixed base and opposite angle
I have a situation where I know the cartesian coordinates of the 2 vertices of a triangle that form its base, hence I know the length of the base and this is fixed.
I also know the angle opposite the base and this is also fixed.
Now what I want to do is figure out how to compute all possible positions for the third vertex.
My maths is rusty, I reverted to drawing lots of pictures and with the help of some tracing paper I believe that the set of all possible vertices that satisfies the fixed base and opposite angle prescribes a circle or possibly some sort of ellipse, my drawings are too rough to discern which.
I started with a simple case of an equilateral triangle, with a base length of two, i.e. the 3rd vertex is directly above the x origin, 0, base runs from -1 to 1 along the x-axis
then i started drawing other triangles that had that same base, -1 to 1 and the same opposite angle of 60 degrees or pi/3 depending on your taste
now i need to take it to the next step and compute the x and y coordinates for all possible positions of that opposite vertex.
struggling with the maths, do i use the sin rule, i.e. sin a / = sin b / B and so on, or do I need to break it down into right angle triangles and then just use something along the lines of a 2 + b 2 = c 2
ultimately, I intend to plot the line that represents all the possible vertex positions but i have to figure out the mathematical relationship between that and the facts, namely,
base is fixed running from (-1,0) to (1,0) angle opposite the base is 60 deg
I then need to extend to arbitrary bases and opposite angles, but thought starting with a nice simple one might be a good stepping stone.