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Have to agree with EP......even worse is that you already signed off.....

I don't think there is a solution for this using the "factorial function"!. However, there is a solution using the "Gamma function". It gives the following solution:

A*B*C =A! + B! + C!, solve for A, B, C

where: A ≈ 4.39056.... and B = -53/10 ≈ -5.30000       and C = -17/10 ≈ -1.70000

Note: This is courtesy of: Mathematica 11 Home Edition.

Seriously???    TEN questions in a single post?   No attempt on your part to answer ANY of these?   We are not here to give answers to your homework or test questions!   

  TRY to answer them ....THEN post a single question at a time....

Yes, you understood it correctly, that's exactly what I was looking for... thank you!! 

If I understand this.....you  are  looking for   " - - "   ????

Looking at the right side

(1/2) [ sin pi/2 - sin pi/6 ]  =

(1/2) [ 1 - 1/2]  =

(1/2) (1/2)  =   1/4

So

" - - "  pi/3 * sin pi/6   = 1/4

" - - " pi/3 * (1/2) = 1/4           multiply both sides by 2

" - - " (pi/3)  =   1/2

The " - - " is    "cos "

Thank you!! 

2x > 3x - 3  ⇒  2x + 3 > 3x

3x - a > -6 ⇒   3x > a - 6

So....we have the combined inequality.....

a - 6 < 3x < 2x + 3          and when x  = 2

a - 6 < 3(2) < 2(2) + 3          

a - 6 < 6 < 7          add 6 to each term

a < 12 < 13

So  a  =  [ 1, 11] =   11 positive integers

We don't need the "squared identities," here...the double-angle identities serve better....

sin^2xcos^2x  = 

(sinx cosx)^2  =                                                                                

[  (1/2) sin 2x] ^2  =                                                         

 (1/4) (sin 2x)^2                       

Note, Cupcake.......     cos 4x  =   1 - 2[sin 2x]^2

                                    2[sin 2x ]^2  = 1 - cos(4x)

                                     (sin 2x)^2  =  [ 1 - cos(4x)] / 2   =  (1/2)[ 1 - cos (4x) ]

So we have

(1/4) (1/2)  (1 - cos(4x) ]    =

(1/8) [ 1 - cos (4x) ]  =

[ 1 - cos (4x) ]

____________

         8

im not quite sure where you got the 64.5? Your equation looks a bit different from the example.

https://web2.0calc.com/questions/help_66866#r1

1 - Let the number of months when they equalize =m
1000 - 60m =600 - 20m, solve for m.
Move the terms with m to the LHS:
-60m + 20m =600 - 1000
-40m  =  -400
Divide both sides by -40:
m = -400 / -40
m = 10 - months when the 2 boys will owe the same amount.

2 - 1000 - [10 x 60] =$400 - This is the amount they both owe after 10 months.

3 - 6 x $60 =$360 - Ian's claim in not true. It will take him: $400 / $60 = 6 2/3 = ~ 7 months to pay off the remaining balance of $400.

This a right tringle with two 45 degree angles    side a   and 7 are the same   a= 7

Pythagorean theorem   s₁^2  + s₂^2 = hypotenuse^2    ( b is the hypotenuse)

                                       7^2 + 7^2 = b^2

                                       49 + 49 = b^2

                                           98 = b^2        rewrite   b^2 = 98    b =sqrt 98 = sqrt( 49 *2) = 7 sqrt 2

Thank you!

I don't think you know what your doing evalete 

whats 2*2

That means 2 times 2.  On this board, an asterisk is sometimes used as a multiplication symbol, especially where using an x as the multiplication symbol could get confused with an "x" used to represent an unknown in an equation. 

x

Determine algebraically and state in how many months the two boys will owe the same amount. State the amount they will owe at this time.  DO NOT LISTEN TO THE ALGEBRAIC PART   

You could make a table, containing each person's amount owed. 

Month        IAN amount owed    KEN amount owed

   0                    1,000                        600

   1                       940                        580

   2                       880                        560

   3                       820                        540 

Continue subtracting payments until you get to the month where the two owed balances are the same.

.

The period is 12 because there are 12 months in a year.

The lowest temp is  40 degrees in January

The highest is 89 in july    (6 months later)

The average of these is   (40+89)/2 = 129/2 = 64.5

89-64.5 = 24.5

y=-24.3cos(nx)

12=2pi/n

n=12/(2pi) = 6/pi

\(y= -24.5 cos \frac{\pi x}{6}+64.5 \qquad\text{The angle is in radians}\\\)

Here is the graph

Oh dear this is meant to be a sinusoidal graph.

The sine curve is just this with a phase shift of a quarter of a period (3 months)

\(y= -24.5 cos \frac{\pi x}{6}+64.5 \qquad\text{The angle is in radians}\\ y= -24.5 sin \frac{\pi }{6}(x+3)+64.5 \\ or\\ y= 24.5 sin \frac{\pi }{6}(x-3)+64.5 \)

Here are the graphs

I think   3!*2 = 12 ways which takes care of rotation  BUT

to take of care of reflection maybe divide by 2 

So maybe 6 ways.

I am not sure though.....

1.) Let \(f(x)=\left\lfloor\left(-\frac58\right)^x\right\rfloor\)   be a function that is defined for all values of x in \([0,\infty)\) such that f(x) is a real number. How many distinct values exist in the range of f(x)?

The real solutions     (-5/8)^x    are  always going to be between -1 and 1 not inclusive   (when x is positive)

so the floor function can only be  -2, -1  and 0

Hence you can answer the actual question.

2) 

f(2)=2a+3=5

2a=2

a=1

f(0)=ab=5

1*b=5

b=5

f(-2)=b*-2+c=-10

-2b+c=-10

-2*5+c=-10

-10+c=-10

c=0

a+b+c =       well you can add up single digit numbers I expect.