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The 2R comes from the standard proof of the sine rule.

Draw your triangle ABC with its circumscribing circle, radius R, centre O.

With the usual notation, angle BOC = 2A, so if you drop a perpendicular from O to BC you get a rt-angled triangle with

two sides R and a/2 and an angle A, from which a/2R = sin(A) and a/sin(A) = 2R.

Repeat for B and C and equate the three.

Here's an alternative answer to the original question.

LHS,

 $\displaystyle \sin(C)=2\sin(C/2)\cos(C/2)=2\sin(90-(A+B)/2)\cos(90-(A+B)/2)\\ =2\cos((A+B)/2)\sin((A+B)/2) \dots\dots\dots(1)$

RHS

$\displaystyle \frac{\sin(A)+\sin(B)}{\cos(A)+\cos(B)}=\frac{2\sin((A+B)/2)\cos((A-B)/2)}{2\cos((A+B)/2)\cos((A-B)/2)}=\frac{\sin((A+B)/2)}{\cos((A+B)/2)}\dots\dots(2)$

Equate (1) and (2), cancel and cross multiply,

$\displaystyle \cos^{2}((A+B)/2)=1/2,\\\text{so }\cos((A+B)/2) = 1/\sqrt{2}\\\text{so }(A+B)/2=45\text{ deg}\\A+B=90\text{ deg}\\C=90\text{ deg}.$

Tiggsy.

Here's one solution:

(Of course, you don't have to use base 60; any base would do - but I only thought of this after doing the above!). 

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Here's the picture.

$\text{All pairs of positions }(m,n) \text{ of the jokers are equally likely}\\ P[0]=P[|m-n|=1] =\dfrac{53}{\dbinom{54}{2}}=\dfrac{2}{27}\\ P[1] = P[|m-n|=2] = \dfrac{52}{\dbinom{54}{2}} = \dfrac{104}{1431}\\ P[k] = P|m-n|=k+1 = \dfrac{54-k-1}{\dbinom{54}{2}}$

$\text{The expected value of the number of cards between jokers is}\\ \sum \limits_{k=1}^{52}\dfrac{54-k-1}{\dbinom{54}{2}}k = \\ \dfrac{1}{1431}\left(\sum \limits_{k=1}^{52} 53k-k^2 \right) = \\ \dfrac{1}{1431}\left(\dfrac{53}{2}(52)(53) - \dfrac 1 6 (52)(53)(105) \right) = \dfrac{52}{3}$

thanks!

$\text{For this problem they probably just want you to list out the 4 series elements and add them}\\ 100(1 + (-4) + 16 + (-64)) = 100(-51) = -5100$

GOOD WORK !!    

I added to your graph :)

Yep. Here is the link to the graph: https://www.desmos.com/calculator/bvpzoqrgnu

So what is the polynomial?  Did you graph it using desmos to check your answer?

squares that end in 6 must be of the form

$(10n\pm4)^2$     where n is an interger 

$(10n\pm4)^2\\ =100n^2\pm80n+16\\ =100n^2+10\pm80n+6\\ =100n^2+10(1\pm8n)+6\\ \text{8n can end in any even number so }1\pm8n \text { can end in any odd number}\\ \text{So the 10s place value can have any odd digit.}$

I've figured out the answer to the question. I just had to do some substitution with the equations. The x-intercepts are (-4,0), (0,0) and (1.5,0). =)

Thanks! The amazing drawing really helped my understand!

All righty then ... show your work or explanation

Start with the original price                                                598.00

Gonna be discounted 25% so how much is that?              598.00 x 0.25 =      149.50

What will the price be after reducing it by 149.50?            598.00 – 149.50 =  448.50

Wait, we're not done. Uncle gets 8% tax. How much?        448.50 x 0.08 =         35.88

So add that tax to the price.  Sigh, always tax.                  448.50 + 35.88 =     484.38

Finally, we got to the bottom line, 484.38 

We get to pay it off over six months, though.

Each payment will be a sixth of the total.                           484.38 x 1/6 =            80.73

The payment each month is $80.73. 

Welcome to the world of high finance. 

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n=1; a=((n+1)^2 / (n+23));printa, n; n++; if(n<500, goto1, 0)

The 4 numbers are: 21, 98, 219, 461

The largest n = 461 

The squre root has to be a 4 or 6  for the square to end in 6

4         16

6             36

14          196

16        256

24     576

 26       676

34            1156

36           1296

44        1936

46        2116                             So I'd say    1  3  9   5  or   7    for the tens' digit.....

Third one:   using negative z-score table

 -.25   S.D.  below the mean represent approx 40%

-.25 x 4.3 =1.075   gm    below the mean       Mean = 120     120 -1.075 = 118.925 gm

Wait a minute...... think this is pre calculus!   HaHa.....

Anyway.....

First one:   29 years is 2 standars deviations fro mean     and 37 is 2 standard deviations ABOVE mean......

  Using _ and + z-score table          2 standard devs below is .0228     2 standard deviations above from the + z-score table is .9772

                                                                        2.28%                                                   97.72%         The area between these two is the amount we are looking to.....               97.72 - 2.28 = 95.44%          95.44% x 400=381.76 ~~ 382 members

Thanks

You didn't offend. I am just stating that your title is incorrect.

It is better if titles reflect the content of the question.

And the content of this questions is

Statistics    OR    Normal distributions     OR       Using Z scores.

Personally I think   'Using Z scores with normal distributions'    would be a very good title.

16^3 x 4     16^2 x 10  + 16^1 x 7   +   16^0 x ?  = multiple of 17

?= units digit = 1

I apologize. I thought this site may be able to help. I did not intend to offend.

Yep, that is what Tertre means :)

That is rather difficult to decipher but have you worked it out?