This is because
is a multivalued function and is not invertible over its entire domain. We restrict its domain to the interval
, which gives one complete branch of values (or one period).
and thus
is outside the domain.
A different way to go about this is to find the value of
first, then compute the inverse tangent of that result. But finding the trigonometric values of multiples of
is somewhat tricky and perhaps more work than is needed.
Instead, we can use a trigonometric identity to find the value of
whenever its argument falls outside the "standard" branch.
We know that
(because
is
-periodic), so
. And now the function can be inverted, so that
This is a rather famous probability problem.
The easiest way to solve this is to calculate the probability that you WON'T roll a "double 6" (or a twelve) each time you roll the dice. There are 36 ways in which dice rols can appear and only one is a twelve. So, for one roll, the probability that you will NOT get a twelve is (35/36)^n where 35/36 is about .97222222 and n would equal 1 for the first trial. So for your first roll the odds that you WON'T get a 12 is .97222222.
For the second roll we calculate (35/36) to the second power or (35/36)^2 which equals about .945216.
When we get to the 24th roll we calculate (.97222222)^24 which equals 0.508596.
For the 25th roll, we calculate (.97222222)^25 which equals 0.494468. For the first time we have reached a probability which is lower than 50 per cent. That is to say, after 25 rolls, we have reached a point in which the probability is less than 50 per cent that we will NOT roll a twelve.
To phrase this more clearly, after 25 rolls we reach a point where the probability is greater then 50 per cent that you will roll a 12 at least once.
Please go to this page 1728.com/puzzle3.htm and look at puzzle 48. (The last puzzle on the page). An intersting story associated with this probability problem is that in 1952, a gambler named Fat the Butch bet someone $1,000 that he could roll a 12 after 21 throws. (He miscalculated the odds [as we know you need 25 throws] and after several HOURS, he lost $49,000!!!)
Please go that page and it has a link to the Fat the Butch story.
Answer:
Sorry ido t k new this please do t rerouted if you do ido t care
Answer:
∠E = ∠F = 41°
Step-by-step explanation:
The sum of the 3 angles in a triangle = 180°
Sum the 3 given angles and equate to 180
x + x + 98 = 180 ( subtract 98 from both sides )
2x = 82 ( divide both sides by 2 )
x = 41
Hence
∠E = ∠F = 41°
