All Answers 358468 Answers

If you want to find the sin(60°) without using a calculator, draw a right triangle with angles of 60°, 30°, and 90°.

From geometny, it can be shown that the length of the side opposite the 60° angle is √3, the side adjacent to the 60° angle is 1, and the hypotenuse is 2 (or multiples of these numbers).

Sin is defined to be the length of the opposite side divided by the length of the hypotenuse:  sin  =  opp / hyp.

sin(60°)  =  √3 / 2

To find the average (the mean) of two numbers, add them together and divide by 2.

The compound interest formula is:  A  = P(1 + r/n)^(n·t)

where          A = final amount          P = amount invested = 100.00        r = rate (as a decimal) = .0217

                   n = number of times compounded per year = 365          t = number of years = 2

Substituting:  A  =  100.00(1 + .0217/365)^(365·2)     --->     A  =  104.44

[I use a dot '.' where you use a comma ','.]

I made some errors in my previous analysis.  A more general approach is derived and used here:

So, in question 3.7 the result is a reduction in speed and in question 3.8 an increase in speed.

(I've used δa for acceleration, but should probably have just used a.  I had δv = aδt in mind .  This doesn't affect the analysis though.)

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You can put this straight into the on-site calculator......

Maybe, it is you that works in the forensic department.  

Could this possibly be  log 8 ???

I did mine "on the fly"....Melody's approach is actually "better"

Thanks Chris    

lo8 ???????

$$\\atan(\frac{1}{4})=A-B\qquad (1)\\\\
atan(\frac{1}{3})=A+B\qquad(2)\\\\
(1)+(2)\\\\
atan(\frac{1}{4})+atan(\frac{1}{3})=2A\\\\
tan[atan(\frac{1}{4})+atan(\frac{1}{3})]=tan[2A]\\\\$$

$$$Now I am going to use the identity$\\\\
\boxed{tan(\theta_1+\theta_2)=\frac{tan(\theta_1)+tan(\theta_2)}{1-tan(\theta_1)tan(\theta_2)}}\\\\\\
tan(2A)\\\\
=tan[atan(\frac{1}{4})+atan(\frac{1}{3})]\\\\
=\frac{\frac{1}{4}+\frac{1}{3}}{1-\frac{1}{4}\times\frac{1}{3}}\\\\
=\frac{\frac{7}{12}}{1-\frac{1}{12}}\\\\
=\frac{7}{12}\div \frac{11}{12}\\\\
=\frac{7}{12}\times \frac{12}{11}\\\\
=\frac{7}{11}\\\\
$This answer is exact$$$

Using the tangent inverse

tan^-1(1/4) = (a - b) = about 14.04°  ......  and tan^-1(1/3) = (a +b) = about 18.43°

So

(a -b) + (a+b) = (14.04 + 18.43)° →  

2a = about 32.47°

So

tan(2a) = tan(32.47°) = about .636363= 7/11

1 raised to any power is just.........  1   !!!!

I'am very glad because I can ask you melody and alan you always help me I'm very Grateful 

so in addition to the mistake which Alan mentioned it also I was thinking that ' t ' is the last second of motion 

so I said 10 s .. but now you told me that ' t ' is the period which 5 s ... 

thank you 

Thanks Alan,

You've given me lots to think about.  Thank you.   

In error - see below.

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Thanks Heureka - much appreciated.

Very crafty the way you did that one geno !!!!

2p+15=39       subtract 15 from both sides

2p = 24          divide both sides by 2

p = 12

And there you are !!!!

Adding 4x to both sides, we just have  y = 4x........ which is just an equation of a line through the origin with a slope of "4."   Thus, there are infinite ordered pairs that satisfy this equation.

For question 3.7:

Not quite right - see below.

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Hi Melody,

use \limits.

Example:

\sum \limits_{n=1}^{\infty} : $$\sum \limits_{n=1}^{\infty}$$

Mekody.....I think this questioner wants to know how many different "PIN" numbers can be created by using "FACTORIALS."  (not, Victorials !!!)

Of course, we would have to know the specifics, but here's a simple example:

Suppose we hade a 6 character "PIN" number consisting of  3 letters and  3 digits, and we allowed repeats. So the total number of possible "PINs" could be formed thusly:

Choose any 3 "slots" to put the letters in, and in each of these slots, we would have 26 choices for the selection of a letter. And for each of these arrangements, we would have 100 possible digit combinations to fill the other 3 "slots" (000-999).

So the total number of "PINs" is just

C(6,3)*26³ *100 = [6! / (6 - 3)! * 3!] * 26³ * 100 = 35,152,000 possible "PINs"

[I think this is correct !!!]

What is 19 1/6 - 4 1/2 - 4 1/4 ?

$$\begin{array}{rcl}
19 \frac{1} {6} - 4 \frac{1} {2} - 4 \frac{1} {4} \\
&=& 19+ \frac{1} {6} - 4 -\frac{1} {2} - 4 -\frac{1} {4} \\\\
&=& 19-4-4 +\frac{1} {6} -\frac{1} {2} -\frac{1} {4} \\\\
&=& 11 +\frac{1} {6} -\frac{1} {2} *\frac{3}{3} -\frac{1} {4} \\\\

&=& 11 +\frac{1-3} {6} -\frac{1} {4} \\\\

&=& 11 -\frac{2} {6} -\frac{1} {4} \\\\
&=& 11 -\frac{1} {3} -\frac{1} {4} \\ \\
&=& 11 -(\frac{1} {3} + \frac{1} {4}) \\ \\
&=& 11 -(\frac{1} {3}*\frac{4}{4} + \frac{1} {4}*\frac{3}{3} ) \\ \\
&=& 11 -\frac{7} {12}\\ \\
&=& 10+1 -\frac{7} {12}\\ \\
&=& 10+\frac{12}{12} -\frac{7} {12}\\ \\
&=& 10 +\frac{12-7} {12}\\ \\
&=& 10 +\frac{5} {12}\\ \\
&=& 10 \frac{5} {12}\\ \\
\end{array}$$

use the site calc on the home page

enter like this      

(19+ 1/6) - (4 +1/2) - (4 +1/4)