CPhill

sec2x = sec^2 x/2-sec^2x

1 /cos2x =  sec^2x/ [2- sec^2x]    and we can write

cos2x /1 =  [2 - sec^2x] / sec^2x

cos2x = 2/sec^2x   - sec^2x/sec^2x

cos2x = 2/sec^2x - 1

cos2x = [2/1] / [1/cos^2x] -1

cos2x = 2cos^2x - 1        which is true

Very nice, Melody........the easiest way of doing things is always the best !!!!!

  We have something like this :

                                  A

                               3

                           D               E

                       4

                 B                                       C

Since DE is parallel to BC, by Euclid........AD/DB  = AE/ EC

And AE  =  9 - EC   .....so we have.....

3 / 4   =  [9 - EC] / EC       cross-multiply

3EC  = 4 [9 - EC]

3EC  = 36 - 4EC

7EC  = 36       divide both sides by 7

EC = 36/7  = about  5.14 ft ....So E should be about 5.14 ft  from C  on AC

Proof :

Note = AE  = 9 - EC  = 9 - 36/7  =  27/7

So

AD/DB = AE/EC

3/4  = [27/7] / [36/7]

3/4  = 27/26

3/4  = 3/4

Note : 

Area of triangle  AOB = (1/2)c * r

Area of triangle AOC = (1/2)b * r

Area of triangle BOC  = (1/2)a * r

So....areae of triangle ABC  = " K"   =

(1/2)c*r  + (1/2)b*r  + (1/2)a *r  =

(1/2)r [ a + b + c ]  =

r  [(a + b + c)  / 2 ]  =

r * s

These will occur in pairs......

The first will occur at  C ( 4n -2, 2)   and the second will occur at C(4n - 1, 2)

So......there are 25 pairs of these from C(2, 2) to C(100, 2)  =  50 integers

Proof of the first :

[ 4n - 2] ! /   ( [4n - 2 - 2]! * 2! ]  =

[4n -2] ! / [(4n-4)! *2! ]  =

[4n -2] * [4n - 3] / 2 =

[4n -2]/2 * [4n -3]  =

[2n -1] * [4n -3]  ..........this is the product of two odds which always  = an  odd

Proof of the second

[4n - 1]!/[ (4n -1 -2)! * 2!] =

[ 4n -1]! / [(4n -3)! 2!]  =

[4n -1][4n-2] / 2  =

[4n -2] /  2  * [4n -1]  =

[2n -1] [4n -1]   =   odd

The volume is given  by  [pi * area of the base * height] / 3 =    [ pi *10^2 * 6] / 3   = about 200pi yd^3  = about 628.32 yds ^3

Just remember, Gamemaster.......we perform multiplication/division from left to right - as we encounter them - before addition and subtraction

1) -3 + (-1) * 2 =  -3 + -2  = -5

2) -10 - 1 / 1 =   -10 - 1  =  -11

3) -6 + (-4) / (-4) =  -6 + 1    =  -5

4) (-6.3) - (-7.46) =   -6.3 + 7.46 = 1.16

5) -7 * 6 * -8 =  -42 * -8   =  336

6) ½ * 1.3 =   .5 * 1.3 =  0.65

6)   could also be worked as     1/2 * [1 + 3/10]  = 1/2 *13/10  = 13/20  

[3/4](x+8)=9   multiply both sides by   4/3

x + 8  =  12    subtract 8 from both sides

x = 4

6.1561196*10^36

The "36" indicates that the decimal point will move 36 places to the right

So.....you would end up with  8 non-zero digits followed by 29 zeros

a.   If the wind is blowing "from" N27E......its actual direction is  243° [ in terms of normal degree  measure]

The plane is traveling in the intended direction of 153° [in  terms of normal degree measure]

To get the  the x ooordinate of resultant vector, we have

145cos153 +  23cos243 =  -139.64

To get the y coordinate of the resulting vector, we have

145sin153 + 23sin243  =  45.36

So...the resuting magnitude = √[ (-139.64)^2 + (45.36)^2]  =   146.8 mph

And the resulting direction =

arctan [ 45.36/-139.64] + 180° = 162°   =   N72W