Melody

I have left many out,

I was thinking that they had to add to 10 , not to a multiple of 10.

You are very welcome :)

Thanks, I'd love to come to South Africa one day  :)

The actual curve when simplified is   y= 2sin(4x)

the simplified differential is    y'=8cos(4x)

the second differential is y''= -32sin(4x)

The minimum will be when  y'=0  and   y">0

y'=0

8cos(4x)=0

cos(4x)=0

4x=90, 270

x=22.5  ,    67.5 degrees

when x=22.5    y-32*sin90 = -1     Maximum

when x=67.5     y"=-32*sin270 = -32*-1 = 32    Minimum.

\(16sinxcos^3x-8sinxcosx\\ =8sinxcosx(2cos^2x-1)\\ =4*2sinxcosx(cos^2x+cos^2x-1)\\ =4*sin(2x)(cos^2x+1-sin^2x-1)\\ =4sin(2x)(cos^2x-sin^2x)\\ =2*2sin(2x)cos(2x)\\ =2sin(4x)\)

This is a sin curve 

it has a amplitude of 2

y=0 is the midline  so     -2<=y<=2

the wavelength will be   360/4 = 90 degrees

90/4= 22.5 degrees

It will pass through

(0,0),    (22.5,2),   (45,0),    (67.5, -2)  and   (90,0)

The minimum value is -2  and this is when x = 67.5 degrees

Here is the graph.

LaTex:

16sinxcos^3x-8sinxcosx\\
=8sinxcosx(2cos^2x-1)\\
=4*2sinxcosx(cos^2x+cos^2x-1)\\
=4*sin(2x)(cos^2x+1-sin^2x-1)\\
=4sin(2x)(cos^2x-sin^2x)\\
=2*2sin(2x)cos(2x)\\
=2sin(4x)

Do you want the answer in degrees or radians? Edit:  Sorry I see it is in degrees.

I have been surprised that your work has been in degrees today.

These types of questions are usually in radians.

Also,

Normally I would attack this as a calculus problem but i i expect your calculus is not so great so I have attacked it from a non-calculus angle. 

I'll answer properly soon.

i just noted down all the possibilities for the aditions.  

These are not the final numbers but the digits that could be contained within the numbers

910000

820000

811000

730000

721000

711100

640000

631000

622000

621100

611110

550000

541000

532000

531100

522100

521110

511111

442000

441100

433000

432100

431110

422200

422110

421111

333100

332110

331111

322210

322111

and that is it, now you need to work out the number of acceptable permutations for each one

eg

532000  

3*5!/3! = 5!/2 = 60

etc

Yes that is correct and it is easier than what I did.

There are many ways to go about this.

You forgot the plus sign though.   

I am glad I could help :)

I do suggest you watch the video that I put up on another or your posts today.

the plus/minus 360 is easy

sin of any angle is equal to the sin of that angle plus or minus any multiple of 360degrees

so

sin23 = sin(23+360) = sin (23 - 720)   = sin (23 +/- 360n)     Where n is an integer

same goes for any other trig function.

with tan we can go one step further

tan is positive in the 1st and third quadrants.

so

   \(tan\theta=tan(180+\theta) =tan (360+\theta) \qquad \text{where theta is an acute angle}\\ also\\ tan(180-\theta)=tan(360-\theta) \\ \\this \;means\;that\;\\ \text{We know that }tan60^o=\frac{\sqrt3}{2}\\ so\;we \;know\\ atan\frac{\sqrt3}{2}=60\\ \text{but it also equals} \;\;60+180, 60-720,etc, \quad 60+180n\\ atan\frac{\sqrt3}{2}=60\pm180n\\ \text{We only need the plus becasue n is any integer, it can be negative.}\\ atan\frac{\sqrt3}{2}=60+180n\;\;\;degrees\\\)

I hope I haven't made any stupid mistakes   

LaTex:

tan\theta=tan(180+\theta) =tan (360+\theta)  \qquad \text{where theta is an acute angle}\\
also\\  tan(180-\theta)=tan(360-\theta)  \\

\\this \;means\;that\;\\
\text{We know that }tan60^o=\frac{\sqrt3}{2}\\
so\;we \;know\\
atan\frac{\sqrt3}{2}=60\\
\text{but it also equals} \;\;60+180, 60-720,etc, \quad 60+180n\\

atan\frac{\sqrt3}{2}=60\pm180n\\
\text{We only need the plus becasue n is any integer, it can be negative.}\\
atan\frac{\sqrt3}{2}=60+180n\;\;\;degrees\\

so

sin135 = +sin (180-135)    Note:  it is in the secend quadrant fo it is positive

sin 135 = sin 45

Maybe you can try the rest yourself.