Alan

The angle, A, made between the horizontal and a line of slope m is given by A = tan^-1(m) so the angle between your two lines is given by  tan^-1(2) - tan^-1(1/2)                (approx 37°)

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Almost asinus.   You are right with cos and tan, but sin(150°) = sin(30°)   (sin is positive in the second quadrant).

Yes.   sinh(x) = (e^x - e^-x)/2  so  sinh(-x) = - sinh(x)

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This is undefined in mathematics.  

You don't supply the pie chart, but it might look something like this:

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Divide by 100:   8% → 0.08

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On third thoughts, Melody is correct!   Easiest way to see this is to consider only four people, two men, two women around a table in four chairs.  There are clearly only two distinct arrangements if rotations are ignored.  Melody's method gives 1!*2! = 2.  

On second thoughts, the men and women changing seat colours just leads to rotations that are equivalent as far as the placement of individuals are concerned, so I think 576 is probably the better solution!

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The men and women must alternate in position.  Imagine there are 4 red seats and 4 blue ones in alternating positions.  

If the men sit in the red seats there are 4! = 24 ways of arranging them.  For each of these there are 4! ways of seating the women in the blue chairs, so there are 24^2 =  576 ways of doing this.  

There are the same number of ways if the men sit in the blue seats and the women are in the red seats, so there are 2*576 = 1152 ways in total.

The above assumes that a rotation of a given arrangement is different from the arrangement of which it is a rotation.

Do you mean \(\frac{x^2+x\times y}{x^2} \rightarrow 1+\frac{y}{x}\) ?