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Does this help:

 Remember that cos(90-α) is the same as sin(α)  (I've used α for angle HAB)

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Your mixtures can never beat fanta ! Fanta is a pure taste and can not be beaten by street sodas! Go somewhere else! Why don't you try this in Saudi Arabia? " iced tea" it's pretty hot there , might work! Don't go to the holy area( as if they'll let you in) cuz fantas EVERYWHERE!!!!!

I think of a number and multiply it 1000 times by 4. The answer is the same as if I had started with twice the original number and multiplied it by 2 how many times?

$$\\x*4^{1000}=(2x)*2^y\\\\
x*2^{2000}=2x*2^y\\\\
x*2^{2000}=x*2^{y+1}\\\\
y=1999$$

Please use the calculator on the home page :)

The perimeter of a rectangular swimming pool is 86 m and its area is 450 square meters. If the width is represented as w, how do you represent its length?

$$\\43-w=\frac{450}{w}\\\\
43w-w^2=450\\\\
w^2-43w+450=0\\\\
w=\frac{43\pm \sqrt{1849-1800}}{2}\\\\
w=\frac{43\pm 7}{2}\\\\
w=25\;\;or\;\;18$$

Length is 25m   width is   18m

I am interested too Alan.  Could you please discuss this one some more ?

I don't get it ! how do you use the angle HAB for this vector ?

can you draw it ! 

Was there supposed to be a picture?  

You need to be a member to post pics :)

i^2 = -1

i^4 = 1

i^40 = (i^4)^10 = 1^10 = 1

(i = sqrt(-1))

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It is 16910...0 where there are 135 zeros.

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You could also use the double angle formula:

$$\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}$$

If θ = 22.5° then 2θ = 45° and tan(45°) = 1 so you have:  

$$1=\frac{2\tan(22.5)}{1-\tan^2(22.5)}$$

This is a quadratic in tan(22.5°) the positive solution of which is √2 - 1 or 0.4142....

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The angle HAB is given by tan^-1(CG/AG) so the horizontal (x) component of force is 2*sin(HAB) kN, and the vertical (y) component is -2*cos(HAB) kN  (assuming y is positive upwards).

Can you take it from there?

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(10-2)-3=5

$${\mathtt{23}}{\mathtt{\,\times\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.175}}\right)}^{{\mathtt{30}}} = {\mathtt{2\,903.112\: \!289\: \!596\: \!765\: \!595\: \!7}}$$

2903 people

Your question makes no sense however

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{22.5}}^\circ\right)} = {\mathtt{0.414\: \!213\: \!562\: \!373}}$$

So the sales price is $65 and you want the original price.

$$\\80\%\; of\; X = \$65\\\\
80/100*X=65\\\\
0.80*X=65\\\\
X=65/0.80\\\\
X=\$81.25$$

Yes, what about it?   

>=    will work

If you hover over the log key, the other bases show up.

(the differrence in the prices ) divided by (the original price)  * 100  percent

You need to use brackets, I do not know what is divided by what.     

A newspaper

You solve inequalities the same as equations EXCEPT

you turn the sign around IF

1) you swap sides

2) you multiply or divide by a negative number

3) you take the reciprocal of both sides.  (turn them upside down)

That is it :)

http://forbrains.co.uk/free_online_tools/convert_to_roman_numerals?toroman=7000000#