CPhill

I thought I was immune.....however....perhaps I need a booster.....

If this is a right triangle, the sum of the squares of the two shorter sides must equal the square of the longer side....so...

40^2  +  25^2   =  60^2     ???

1600 +  625   =  3600   ???

2225   =  3600  ???

Not true....so.....no right triangle

Gee, Guest....take a Valium......!!!!

Everyone isn't endowed with your superior math abilities.......

However.......until you have some famous theorem named in your honor, we'll continue to try to help the great unwashed.....

Focus  =  ( -1, - 3)   

Directrix  ...   y  =  1

The vertex will  lie  at     (-1 ,  [ -3 + 1] / 2 )   = ( -1, -2/2 ) =    ( -1, -1)

So, p =  -2  ......and we have

4p ( y + 1)  =  (x + 1)^2

4 (-2) ( y + 1)  =  ( x + 1)^2

-8 ( y + 1)  = (x + 1)^2        multiply through by -1/8

y + 1   =  (-1/8)(x + 1)^2      subtract 1 from both sides

y  =  -(1/8)(x + 1)^2   -   1

I believe this is correct......

When  r   =  -1/2    the series sums to    9 / [ 1 -  (-1/2)]  =  9 / (3/2)  =  18/ 3    =   6

Excellent answer, X².....!!!

I might add....another way of remembering the basic Trig identities....

Sine   =  Oh / Heck

Cosine = Another / Hour

Tangent = Of / Algebra        !!!!

Correct  !!!!

The area   =  (1/2)(radius)^2 * ( radian measure of the central angle)  =

(1/2) (60^2) *(pi / 6)   =

(1/2) (3600) * (pi/6)  =

3600pi / 12

300pi  mm^2

Note that   5/7   =  0.7142857     [ the  7142857 is a repeating string ]

Try this  :

 a₂ = 1, a₃  = 1, a₄  = 1,  a₅  = 0, a₆ = 4, a₇ = 2

1/2! + 1/3! + 1/4! + 0/5! + 4/6! + 2/7!   =     5 / 7

I didn''t really have a mathematical way to solve this.....but, note  that

1/2!  + 1/3! + 1/4!  + 1/5! = 43/60 > 5/7

And 

1/2!  + 1/3!   + 1/4!  = 17/24  <  5/7

So......we need that    5/7  - 17/24   =  1/168   =   a₆/6! + a₇/7!

So........it was just a case of  letting a₅ = 0 and then  manipulating a₆  and a₇  to find a solution

Perhaps some other mathematician can find a more elegant way to the solution....!!!!

Using  (a^m)^n   =   a^(m * n), we have

(3^8)^4   =   3^(8 * 4)   =  3^32

I believe so......

Here's another one....

Prove that

Sin (x)  /  n       =    6