Problems

Now for some exercises upon which to practice these ideas. The first three are easy if you apply the correct trigonometric identity. The next eleven problems apply the Seven Wonders of the World, Jensen's Inequality, AM-GM Inequality and/or previous exercises.

  1. If $a^2 + b^2 = 1$ and $m^2 + n^2 = 1$ for real numbers $a,b,m$ and $n$, prove that $\vert am+bn\vert \le 1$.
  2. Solve $3\sin^2\alpha-4\sin^4\alpha - 2 = 0$.
  3. ( 1984 ARML) In triangle $ABC$, $a \ge b \ge c.$ If $\frac{a^3+b^3+c^3} {\sin^3\alpha +\sin^3\beta +\sin^3\gamma} = 7$, compute the maximum possible value for $a$.
  4. $\sin\alpha\sin\beta\sin\gamma\le\frac{3\sqrt{3}}{8}$.
  5. $\csc\alpha\csc\beta\csc\gamma\ge\frac{8\sqrt{3}}{9}$.
  6. $\frac{3}{4}\le \cos^2\alpha + \cos^2\beta + \cos^2\gamma < 3$.
  7. $ \sec^2\alpha +\sec^2\beta+\sec^2\gamma > 3$.
  8. $ \csc^2\alpha +\csc^2\beta+\csc^2\gamma \ge 4$.
  9. $1 < \sin\frac{\alpha}{2} + \sin\frac{\beta}{2} + \sin\frac{\gamma}{2}\le \frac {3}{2}$.
  10. $2 < \cos\frac{\alpha}{2} + \cos\frac{\beta}{2} + \cos\frac{\gamma}{2}\le \frac {3\sqrt{3}}{2}$.
  11. $\tan\frac{\alpha}{2} + \tan\frac{\beta}{2} + \tan\frac{\gamma}{2} \ge \sqrt{3}$.
  12. $\cot\frac{\alpha}{2} + \cot\frac{\beta}{2} + \cot\frac{\gamma}{2} \ge 3\sqrt{3}$.
  13. $\csc\frac{\alpha}{2} + \csc\frac{\beta}{2} + \csc\frac{\gamma}{2} \ge 6$.
  14. $\sec\frac{\alpha}{2} + \sec\frac{\beta}{2} + \sec\frac{\gamma}{2} \ge 2\sqrt{3}$.