[Soln] 2014 Singapore MO (Senior Rd 1) Problem 7

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

(A) \tan 47^\circ + \cos 47^\circ   (B) \cot 47^\circ + \sqrt{2}\sin 47^\circ   (C) \sqrt{2}\cos 47^\circ + \sin 47^\circ   (D) \tan 47^\circ + \cot 47^\circ   (E) \cos 47^\circ + \sqrt{2}\sin 47^\circ

A good math olympiad student should have the graphs of y = \sin xy = \cos x and y = \tan x memorized:

SMO_2014_SrRd1_7_01(image courtesy of xpmath.com)

\cot x 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 \cot x = 1 / \tan x to derive the graph of y = \cot x:

SMO_2014_SrRd1_7_02(image courtesy of xpmath.com)

Now back to the question. Nobody remembers what the value of any trig function of 47^\circ is, so this is an estimation problem. The closest nice angle is 45^\circ, so we should try approximating all trig functions of 47^\circ as trig functions of 45^\circ. For every trig function (denote as f for now to be general), we can estimate as such:

f(47^\circ) \approx f(45^\circ) \pm \varepsilon,

where \varepsilon is some small number, and it is + or – depending on the graph. Now we know that for each trig function the value of \varepsilon 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 \varepsilon‘s compare with each other later if there is a need for a tie-break.)

Using the graphs above, we can write:

\begin{aligned} \sin 47^\circ &\approx \sin 45^\circ + \varepsilon &&= \frac{1}{\sqrt{2}} + \varepsilon, \\  \cos 47^\circ &\approx \cos 45^\circ - \varepsilon &&= \frac{1}{\sqrt{2}} - \varepsilon, \\  \tan 47^\circ &\approx \tan 45^\circ + \varepsilon &&= 1 + \varepsilon, \\  \cot 47^\circ &\approx \cot 45^\circ - \varepsilon &&= 1 - \varepsilon. \end{aligned}

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

\begin{aligned} &(A): \tan 47^\circ + \cos 47^\circ &&\approx 1 + \frac{1}{\sqrt{2}},&& \\  &(B): \cot 47^\circ + \sqrt{2}\sin 47^\circ &&\approx 2 &&+ (\sqrt{2} - 1)\varepsilon, \\  &(C): \sqrt{2}\cos 47^\circ + \sin 47^\circ &&\approx 1 + \frac{1}{\sqrt{2}} &&+ (1 - \sqrt{2})\varepsilon, \\  &(D): \tan 47^\circ + \cot 47^\circ &&\approx 2,&& \\  &(E): \cos 47^\circ + \sqrt{2}\sin 47^\circ &&\approx \frac{1}{\sqrt{2}} + 1 &&+ (\sqrt{2} - 1)\varepsilon. \end{aligned}

The term with \varepsilon 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 \cot 47^\circ, it boils down to whether \sqrt{2} \sin 47^\circ or \tan 47^\circ 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:

\begin{aligned} &\sqrt{2} \sin 47^\circ &&> \tan 47^\circ, \\  \Leftrightarrow &\sqrt{2} \sin 47^\circ &&> \frac{\sin 47^\circ}{\cos 47^\circ}, \\  \Leftrightarrow &\sqrt{2} &&> \frac{1}{\cos 47^\circ}, \\  \Leftrightarrow &\cos 47^\circ &&> \frac{1}{\sqrt{2}}, \end{aligned}

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

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