All Answers 356076 Answers

I think you should firstly tell them that addition and subtraction are EQUALLY IMPORTANT 

They must be done from left to right.

ALSO

The minus sign is 'glued' to the 3.  They are in fact adding minus 3.  If the terms are rearranged the minus must always be in front of the 3

Thank CPhill for the correction and the detailed description of the correct solution.
I was a bit frivolous with the division by cos (x).
That will not happen to me.
Greeting asinus: - )

I live in the US and we term it PEMDAS, but the principles are still the same......

1. Work  inside the brackets [parentheses], first

2. Perform any exponentiation, next

3. Perform any multiplications/divisions as we encounter them from left to right........this is the thing that usually gives people the most trouble.......many people think that multiplications are performed BEFORE divisions, but this is incorrect......for example:

10 / 2 * 5   =   10 / 10  =  1     [ incorrect ]

10 / 2 * 5   =   5 * 5  =  25   [ correct ]

4. Next, perform any additions / subtractions as we encounter them from left to right

BTW ...using (4) above.......   10 - 3 +  7    =   7 + 7     = 14  

sec^2 x   +   csc ^2  x   =

1 / cos^2 x    +    1/sin^2x     =

[sin^2 x  + cos^2 x ]  /  [ sin^2x * cos^2x]  =

1 / [sin^2x * cos^2x]   =

1/sin^2x  * 1/cos^2x   =

csc^2 x * sec^2 x          [other answers are possible  ]

We don't have an angle "length"....I think you wantt the length of the adjacent side

If one angle = 8.5° and the opposite side = 45, the other adjacent  [acute]  angle = 90 - 8.5 = 81.5°

And using the Law of Sines, we can find the adjacent side, x, opposite this angle thusly :

x / sin (81.5°)  =  45 / sin (8.5 °)      multiply both sides by sin (81.5°)

x  = 45* sin(81.5°) /  sin (8.5°)   = about  301.102   = Answer "c"

And the hypotenuse   = √ [ 45^2 + 301.102^2 ]    = about  304.446

Note that  :

sin (8x - 6x)   = sin 8x cos 6x  -  cos 8x sin 6x     ....so......

-sin (8x - 6x) =  cos 8x  sin 6x  - sin 8x cos 6x   ...... and this implies that.......

-sin (8x - 6x)  = -0.85

sin (2x)  = 0.85

Using the sine inverse 

arcsin (0.85)   = x =  about 58.21°

So.....dividing this angle  by 2,   x = about  29.105°

Here's a graph of the solution :  https://www.desmos.com/calculator/u95uehqy1i

1.2 x  10^-15   =

.0000000000000012

We need to be careful about "dropping a solution" when solving this kind of problem.

Dividing away cos x will eliminate possible solutions........

We have

cosx / sinx = 2cosx      rearrange as

2cosx - cosx / sin x  = 0       factor out cos x

cosx ( 2  - 1/sinx) = 0

cos x ( 2 - csc x )   = 0        set both factors to 0

cos x   = 0     and this happens at   pi/2 + n * pi      where n i s an integer

2 - csc x   = 0     

2 = csc x        this is the same as .......      1/2 = sin x

And this happens at  pi/6  + n* 2pi       and at     5pi/6 + n *2pi       where n is an integer

I also believe that the Guest's answer is correct........to see why........note that 2 MUST be included in the sum......if not, the sum of any other  three primes less than 200 would be odd since all the other primes less than 200 -   besides 2  -  are all odd.......

Thus......we're looking for the largest product of two primes that sum to 198.....this will occur when these two primes are nearest 198/2, i.e.,  99.......the two primes that fit this bill are 101 and 97

Current Population=CP

Future Population =FP

FP=CP x 1.01^x

2 = 1 x 1.01^x take the log of both sides

x =Log(2) / Log(1.01)

x=69.66 years that it will take the population to double @ 1%

1.26262626262... × 10^-20

270 / 5 =54, therefore choose 2 numbers below this and 2 numbers above it, that is:

52 + 53 + 54 + 55 + 56 =270

The best that I can come up with is the following:

2 + 101 + 97 =200

2 x 101 x 97 =19,594

Which is equivalent  to (6^2) (6^4)=6^(2+4)=6^6. The answer is "B"

Hello Guest!

How do you solve cos(x) / sin(x)=2cos(x)

cos(x) / sin(x) = 2 cos(x)                    [ / cos(x)

1 / sin(x) = 2                                      [ reciprocal

sin(x) = 0.5

x = arcsin(0.5)

x = 30°  90°   150°

Put to the test:

cos(30°) / sin(30°) = 2 cos(30°) 

0.866 / 0.5 = 2 * 0.866

cos(90°) / sin(90°) = 2 cos(90°) 

0 / 1 = 2 * 0

cos(150°) / sin(150°) = 2 cos(150°) 

(-0866) / 0.5 = 2 *(-0.866)

Greeting asinus :- )

   !

over 9000!!!!

 formula for PV for annuity due of 8000 12% per annum 240 monthly payments

PMT={PV. R.{[1 + R]^N / [1 + R]^N - 1}} / [1 + R], Where R=Interest rate per period, N=number of periods, PMT=periodic payment, PV=Present value.

PMT={8,000.0.12/12{[1 + 0.12/12]^240 / [1 + 0.12/12]^240 - 1}} / (1 + 0.12/12)

PMT=$87.21

Using the fact that    a^m * a^n   = a^(m + n), we have

(6^3) × (6^-2)   =  

6^ (3 -2)   = 

6^1   =  

6

 8.3005 × 10^-6  = 

0.0000083005

The absolute change  is   4400  -  3550   = 850 people

The percent change is given by :

(  [ 4400 - 3550 ] / 3550  * 100 ) %   =  about 23.94%

the answer is 7036

The area would be given by :

[ a + 3b] *  [ a - 3b ] =

a^2 - 9b^2

The form of a quadratic is   f(x) = ax^2 + bx + c .......and we have that

24 = a(1)^2  + b(1)  + c    →      24   =  a + b + c         (1)

0 =   a(4)^2  + b(4) + c    →         0  = 16a + 4b + c     (2)

60 = a(7)^2 + b(7) + c   →         60  =  49a +  7b + c    (3)

Multiply  (1) by -1 and add it to  (2)    

-24  = 15a + 3b      (4)

Multiply (1) by -1 and add it to (3)

36 =  48a + 6b   (5)

Multiply   (4) by -2 and add to (5)

84   = 18a    divide both sides by 18

84/18 = 14/3   = a

Substituting this into (4)  to find b, we have

-24  = 15(14/3)  + 3b

-72  = 15*14 + 9b

-72  = 210 + 9b

-282  = 9b     divide both sided by 9

-282/9   = -94/3  = b

And using (1)  to find  "c"  

14/3 -  94/3 + c   = 24

-80/3  + c = 24

c = 72/3 + 80/3 =   152/3

So....the quadratic is

y f(x) = (14/3)x^2 - (94/3)x + 152/3

And f(-1)   =    14/3  + 94/3 + 152/3   =   260/3   

Here's a graph : https://www.desmos.com/calculator/jjflwmz37w

Here's the graph : 

This is best done as a "proof by contradiction"

Let us suppose that ∛4  can be expressed as the rational  a/b   where a, b are relatively prime integers -  [ have no common factors ]

So we have 

∛4  = a/b        cube both sides

4   = a^3  / b^3       multiply both sides by  b^3

4b^3   =  a^3        since the left side is even, then a^3 must also be even.....and since the product of an odd integer cubed would also be odd, then "a" itself must be even

Then, since  "a" is even,  and we can write any even  number as  "2m,"  let us write "a"  as "2m"    ....and we have

4b^3   = (2m)^3    simplify

4b^3  = 8m^3       divide both sides by  4

b^3  = 2m^3        note that the right side is even, so this would imply that b^3 is also even

And, as before......if b^3 is even, then so is "b"

But if "a"  is even   and "b"  is even, then they both have a common  factor of 2.......but......this violates our assumption that a and b are relatively prime......so....our assumption that ∛4  can be expressed as the rational  a/b  must be incorrect..... and .this means ∛4  is irrational........

any number to the power zero is  1.