CPhill

3sin(θ/2) = 3/2      divide both sides by 3

sin(θ/2) = 1/2

Now the sine θ = 1/2 when θ =  pi/6 and 5pi/6

So, set θ/2 =pi/6

Multiply both sides by 2

θ = pi/3

And set θ/2 = 5pi/6

Multiply both sides by 2

θ = 10pi/6 = 5pi/3

So...our two answers are pi/3 and 5pi/3 in the given interval

Sorry....but we DON'T need "common denominators" when we multiply fractions.....besides multiplying (-3 or so)  times (-4 or so) couldn't possibly give us 210 !!!.......that answer lacks the "reasonableness test "

Here's the way to do this..........

Convert both fractions to improper ones

-4 + 1/5 = -21/5    and  -3 + 1/3  = -10/3

Now....just multiply them together

(-21/ 5) (-10/3)....however...wait.......here's a trick to make it easier.......a negative times a negative will produce a positive result - we can "drop" the negative signs - and we can also "swap" denominators and "cancel" before we multiply....so we have...

(21/3) (10/5) = (7)(2) = 14 !!!

And that's definitely NOT 210 !!!

Assuming that $300 and $50 are the respective cost(s) of the premium and discount tickets, if we didn't buy any premium tickets, we could afford to buy 12 discount tickets at $50 because 12($50) = $600

The number of total sets possible by choosing any 6 people from 15 = C(15,6) = 5005

But the only ones we're intereseted in are the ones where we choose 4 men from 8 and 2 women from 7 = C(8,4) * C(7,2)  = 1470

So.....1470/5005 = about 29.3%

Thanks for that website reference, Melody.....I can never quite summon the words to explain how to do this - even though it's easy !!! As they say....the pics on the website are worth 1000 words !!! (Well....I don't actually know if anyone used those exact words.....)

S = the semi-perimeter = (15 + 12+ 10) / 2  = 18.5

So we have

A = √[(18.5)*(18.5-15)*(18.5-12)*(18.5-10)]= √[18.5 * 3.5 * 6.5 * 8.5]= 59.8 m²

BTW......this "formula" is known as "Hero's Formula" or "Heron's Formula".....however, many mathematicians believe that it bears the distinct "pawprint" of Archimedes...the proof of it, geometrically, is somewhat involved......something at which Archimedes excelled !!!

(-8)^-6 =

1 / (-8)⁶ =

1 / 262144

The area of a square is just = (side length)²

So we have.........

412cm² = s²

To find the side length, we would take the square root of 412

But, the square root of 412 must be between 20 and 21, because 20² = 400 and 21² = 441

So the side length estimate is between 20 and 21 cm in length

The square root of 121= 11.....and any integer is a rational quantity....

If you are divisible by 9, then you are divisible by 3. Then, the sum of your digits is also divisible by 3.

But you must be even, since you are divisible by 2.

And if you're divisible by 2 and 9, then you are divisible by 18, as well.

So the possibilities are 426, 432, 438, 444 and 448

And 432 is the answer.  No other answers are possible because all of the other numbers lie within 18 integers - positive or negative - from 432.