**2014 SMO (Senior-Rd1) 7.** Find the largest number among the following numbers:

(A) (B) (C) (D) (E)

A good math olympiad student should have the graphs of , and memorized:

*(image courtesy of xpmath.com)*

is not used particularly often in olympiad math so there is no real need to memorise its graph. In any case, one can use the definition to derive the graph of :

*(image courtesy of xpmath.com)*

Now back to the question. Nobody remembers what the value of any trig function of is, so * this is an estimation problem*. The closest nice angle is , so we should try

*. For every trig function (denote as for now to be general), we can estimate as such:*

**approximating all trig functions of as trig functions of**,

where is some small number, and it is + or – depending on the graph. Now we know that for each trig function the value of is going to be different, but * since we are just estimating, we can assume they are all the same for now*. (We may need to think about how the different ‘s compare with each other later if there is a need for a tie-break.)

Using the graphs above, we can write:

We can plug these expressions in to options (A) to (E) to get estimates for them:

* The term with should be negligible so we can focus on comparing the first term*. The prime candidates for being the largest are (B) and (D). Noting that both of them have the term , it boils down to whether or is larger.

* One way to determine which is larger is to assume that one is bigger than the other, then manipulate the equation to a point where it becomes obvious whether the inequality is true or not.* If we assume that (B) is larger than (D), then:

which is clearly false. Hence, our original assumption must be false, meaning that (D) is larger than (B). The answer is **(D)**.