Try this beautiful problem from the American Invitational Mathematics Examination, AIME 2015 based on Trigonometry.

Trigonometry Problem – AIME 2015


With all angles measured in degrees, the product (\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n), where (m) and (n) are integers greater than 1. Find (m+n).

  • is 107
  • is 91
  • is 840
  • cannot be determined from the given information

Key Concepts


Trigonometry

Sequence

Algebra

Check the Answer


But try the problem first…

Answer: is 91.

Source
Suggested Reading

AIME, 2015, Question 13.

Plane Trigonometry by Loney .

Try with Hints


First hint

Let (x = \cos 1^\circ + i \sin 1^\circ). Then from the identity[\sin 1 = \frac{x – \frac{1}{x}}{2i} = \frac{x^2 – 1}{2 i x},]we deduce that (taking absolute values and noticing (|x| = 1))[|2\sin 1| = |x^2 – 1|.]

Second Hint

But because \(\csc\) is the reciprocal of \(\sin\) and because \(\sin z = \sin (180^\circ – z)\), if we let our product be \(M\) then[\frac{1}{M} = \sin 1^\circ \sin 3^\circ \sin 5^\circ \dots \sin 177^\circ \sin 179^\circ][= \frac{1}{2^{90}} |x^2 – 1| |x^6 – 1| |x^{10} – 1| \dots |x^{354} – 1| |x^{358} – 1|]because \(\sin\) is positive in the first and second quadrants.

Final Step

Now, notice that \(x^2, x^6, x^{10}, \dots, x^{358}\) are the roots of \(z^{90} + 1 = 0.\) Hence, we can write \((z – x^2)(z – x^6)\dots (z – x^{358}) = z^{90} + 1\), and so[\frac{1}{M} = \dfrac{1}{2^{90}}|1 – x^2| |1 – x^6| \dots |1 – x^{358}| = \dfrac{1}{2^{90}} |1^{90} + 1| = \dfrac{1}{2^{89}}.]It is easy to see that \(M = 2^{89}\) and that our answer is \(2+89=91\).

Subscribe to Cheenta at Youtube