Tuesday, April 14, 2015

Olympiad Trigonometric Problem $\cos^{48}\dfrac{P}{2}\sin^{23}\dfrac{Q}{2}=\cos^{48}\dfrac{Q}{2}\sin^{23}\dfrac{P}{2}$

Great and professional teaching methodology could definitely help to mold one's thinking and stimulate the inquisitive mind of students. In order to achieve this noble aim, we need good resources, such as the state of the art technologies, professional educators, competitive peers, supportive family members and above all, the challenging and intriguing math problems!

I will present one really intriguing problem here, and let's see how we are going to milk it for all its worth...

In a triangle $PQR$ the following equality holds:


Find the value of $\dfrac{PR}{QR}$.

What is the first thing sprang to mind when you saw this problem? Would you hastily apply the relevant trigonometric identities or formulas that you are so familiar with to solve for the values of $\angle P$, $\angle Q$ and therefore $\angle R$?

Would you ponder for a minute to determine if the method for solving the angles of the triangle from the given expression would work?!?

Can you picture success in your mind with whatever trigonometric formulas that you think could help to simplify the given expression in two variables and then solve for them?

What could we do, if, let us say, we obtained the values for the angles for that triangle $PQR$? Would it be feasible to generate the value for $\dfrac{PR}{QR}$ then?

This sounds like we have too much uncertainties with this suggested method but what other credible alternatives do we have?

Could you coax the students to name, at least one alternative for solving the problem?

Even if they could not think of anything else that is helpful, you, the educator should instead encourage them to go with the current method and see how far they can go with it and when should they call an end to it. There would be no silly mistakes or wasted effort if we learn from our mistakes. As a matter of fact, when we ask our students to try something that we know for sure would not work, and they reached to impasse after they tried, they would get more curious about how one should approach the problem wisely and they would open their mouth and begin to ask! Albert Einstein once said,

The important thing is not to stop questioning. Curiosity has its own reason for existing.”

This goes without saying...

I will post the solution soon but I sincerely hope you would encourage your students to try it out first, it really doesn't matter if they met with failures in their attempts, the only things that matter is they tried!

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