frasinscotland

You are getting confused. They are not positioned at the same place.

i   X  sin(pi/2)  = i   X    1      

                           = i.

But  sin(pi/2) is  just  1 .       So  1    and  i  are not positioned at the same place,far from it.  I think you need to look some more at complex numbers     (they aren't really all that complex ,someone just decided to call them by that name)  and get some practise with them.

First, length of OA is \(\sqrt50\)as it's a radius of the circle. No need to simplify this,soon see why..

Now since line AP lies on the tangent line L ,then AP/OA   = tan 90 deg

So AP/OA = 1

AP/\(\sqrt50 = 1\)

So AP is also \(\sqrt50 \)

Area of triangle is 1/2 base  X height

          = 1/2   X \(\sqrt50*\sqrt50\)

          =  1/2  X 50

          =  25

4th quadrant.

                     Sine positive      All positive

                     Tan positive      Cosine positive                           are the positive values for the 4 quadrants.

I remember it using the acronym '    Acts clockwise.'                                               

No.  It has a constant velocity of  2m/s.   Acceleration means that the  velocity is increasing (  it is changing   ie  NOT constant).   If it is stated that the object is moving with a constant velocity,it cannot be accelerating. 

The difference : Constant velocity ,the object's  velocity stays the same and never changes

                        : acceleration , the object's velocity is continually changing (increasing)

Also note that

if the  constant velocity is given in m/s  

then the acceleration will be given in m/s/s    which is usually written as  m/s^(-2)

Much easier and faster to realise that if cot (x)  = -2  then

tan (x) = -1/2 , construct the triangle and read off the value of sin (x)

----construct the triangle in the 4 th quadrant where the side  of the triangle on the x- axis  has length 2

hypoteuse has length Sqrt5 and opposite side to angle (x) has length  - 1             ie this means that tan (x) = -1/2

Now you can just read off the value by looking at the triangle ,  sin (x) =  - sqrt (1/5)  .  And it's also obvious from the diagram that sin (x) has to be negative.....

If you have a square whose side is 12 inches long,then the area of the square is   12   X   12  square inches  which is

144   square inches.    That's the simplest shape, you can of course have others.

You can factor out in a few ways,if that is the question

realise that a^3 X b^2    =  a x a x a x b x b       so you can have 

                                      = a(a x a x b xb)    = a(a^2b^2)       

or a^2(a x b^2)     etc   etc    .     But in most calculations, a^3b^2 is probably the simplest form.   

The question is

What per cent of  102  is 32   ?      that is

x per cent of  102   is 32         Remember that per cent means  hundreths, also when we say   'of'  we multiply. A half  'of' something  is 1/2    X   something.   So to write 

x  per cent of 102 is 32       as an equation  ,  we write

x/100            X         102    = 32         ( the capital X  is a multiplcation sign,don't confuse with little x )

re-arranging

x/100    = 32/102        (divide both sides by 102)

x = 100  X  (32/102)       =   31.372.

Same method for the other two,try for yourself.

Hi, first of all   ALL quadratic equations have solutions. Lots more to say,but we'll keep it simple.These are an extremely important family of functions lots of reasons ; they even describe the shape of your tv satellite dish!

The ones you'll be seeing will either factorise nicely,or the formula will always get you there.That is

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)A quick way too see if it's going to factorise nicely is to look at the expression inside the square root sign,that is \(b^2-4ac\)

If \(b^2-4ac\)is a positive square number  ie 16,  25,  81 etc    it will always factorise nicely. But if you are having difficulty factorising (it's just lots of practise!)  the formual will always work too.

{If \(b^2-4ac\)is negative,you will not at this stage in your maths career be able to solve the quadratic, but you will not be asked to solve one of these. If you were to look at a graph of a quadratic where b^2 -4ac is negative,you would see that the curve never touches the x -axis. Sometimes,you might be asked to find the maximum or minmum value of one of these curves,using a technique called Completing the Square.} 

Plenty of people on here I'm sure will be happy to help you, just do lots of examples and you will soon get the technique of factorising.

Cantor is spinning in his grave.....  I am willing to stand corrected but -

I think the answer is that there is an infinite number of co-primes that satisfy this. I can write the numerator x  as the product of factors

(a1*a2*a3....an)  and the denominator y  as the product (b1*b2*b3...bm)   where no a's or b's are equal,to guarantee that I have co-primes.  So

x/y =(a1*a2*a3...an) / (b1*b2*b3...bm)              {  n  and m to provide that  x and y don't need to have the same number of factors}             Clearly I can keep choosing more and more a's  and b's to generate more and more co-primes  ie an infinite set.        What about (x+1)/(y+1)  ?     This is

(a1*a2*a3...an  + 1) /(b1*b2*b3....bm  +1)   and we are requiring from these infinite sets that

(a1*a2*a3....an  +1)/(b1*b2*b3.....bm +1)   =(1.1) (a1*a2*a3....an)/(b1*b2*b3....bm) .   Again,every time I reach a solution,I can choose more a's and b's  until I find another one   ad infinitum.   

\(n^5 / n^3 = n^2\)

Is this what you are asking?  Laws of indices,  when you divide you subtract the index of the denominator from the index of

the numerator.                 5-3  =2 

In general

X^n   - X^m   = X^(n-m)

\(2^{x}-2^{x-1} =4 \)now factor out  to get

\(2^{x-1} (2^{1}- 1)=4\)

\(2^{x-1}(1) =4\)

\(2^{x-1}=4 \)  since we know that \(2^{2}=4\)

then x-1  =2    so x =3.

Check  2^3   - 2^2    =   8 - 4     =4