Alan

cosh(x) = (e^x + e^-x)/2

sinh(x) = (e^x - e^-x)/2  so

sinh(3*pi*i) = (e^3*pi*i - e^-3*pi*i)/2

Now e^iy = cos(y) + i*sin(y) so

sinh(3*pi*i) = (cos(3pi)+i*sin(3pi) - cos(3pi)+i*sin(3pi))/2 = i*sin(3pi) = i*0 = 0

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I suspect it's Adobe Flash player as that's what seems to be running the animation.

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I think the sine rule might be the simplest approach here:

h/sin(H) = j/sin(J)

$${\mathtt{h}} = {\frac{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{132}}^\circ\right)}{\mathtt{\,\times\,}}{\mathtt{31}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{21}}^\circ\right)}}} \Rightarrow {\mathtt{h}} = {\mathtt{64.284\: \!458\: \!526\: \!596\: \!557\: \!1}}$$

or h = 64 to the nearest whole number

I've assumed h is opposite angle H, j is opposite angle J etc.

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If you want to enter something like 

$$1\frac{2}{3}$$

then just type (1 + 2/3)

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The firstfour terms of (1 + m)⁶ are 1 + 6m + 6*5/2*m² + 6*5*4/(3*2)*m³

So if m = x/3 

a = 6/3 = 2

b = 6*5/2*(1/3²) = 30/18 = 5/3

c = 6*5*4/(3*2)*(1/3³) = 20/27

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Use the mod function on the calculator here

$${\mathtt{3\,997}} \,{mod}\, {\mathtt{5}} = {\mathtt{2}}$$

2 is the remainder when 3997 is divided by 5

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I think you must mean 855 degrees (not 885 degrees).  Most calculators won't display results as fractions, but you can do this by continuing to subtract 360 degrees from this until you get to 135 degrees (855 - 360 -360 = 135).  sec, like cos,  is negative here (sec = 1/cos).

Imagine a line on a graph from the origin (0, 0) to the point (-1, 1).  This is at angle of 135 degrees to the positive x-axis (45 degrees to the negative x-axis).  

The length of the diagonal of this line is √(1^2 -(-1)^2) = √2

Then sec(135) = √2/(-1) = -√2

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Put the decimal point after the first non-zero digit, then multiply by ten to the power of however many decimal places are needed to complete the number as in your example.  If the number is smaller than 10 then you will need a negative exponent on the ten.  For example:  0.0108 = 1.08*10^(-2)

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$$\frac{1-x^2}{x^{3/4}}=\frac{1}{x^{3/4}}-\frac{x^2}{x^{3/4}}=x^{-\frac{3}{4}}-x^{2-3/4}=x^{-\frac{3}{4}}-x^\frac{5}{4}$$

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