## Category Archives: Trigonometry

### De Moivre’s Theorem application

Question:

If $f_{r}(\alpha)=(\cos{\frac{\alpha}{r^{2}}}+i\sin{\frac{\alpha}{r^{2}}}) \times (\cos{\frac{2\alpha}{r^{2}}}+i\sin{\frac{2\alpha}{r^{2}}}) \ldots (\cos{\frac{\alpha}{r}}+i\sin{\frac{\alpha}{r}})$, then

$\lim_{n \rightarrow \infty}f_{n}{\pi}$ equals

(a) -1

(b) 1

(c) -i

(d) i

Solution.

Using De Moivre’s theorem,

$f_{r}{\alpha}=e^{i\frac{\alpha}{r^{2}}}e^{i\frac{2\alpha}{r^{2}}}\ldots e^{i\frac{\alpha}{r}}$

which in turn equals $e^{(i \frac{\alpha}{r^{2}})(1+2+\ldots+r)}=e^{(i\frac{\alpha}{r^{2}})(\frac{r(r+1)}{2})}=e^{i(\frac{\alpha}{2})(1+\frac{1}{r})}$

Hence, $\lim_{n \rightarrow \infty}f_{n}(\pi)=\lim_{n \rightarrow \infty}e^{(i)(\frac{\pi}{2})(1+\frac{1}{n})}=e^{i(\frac{\pi}{2})}=\cos{\frac{\pi}{2}}+i\sin{\frac{\pi}{2}}=i$.

More complex stuff to be continued in next blog (pun intended) 🙂

Nalin Pithwa

### More trigonometry practice

Problem.

Let ABC be a triangle and $h_{a}$ be the altitude through A. Prove that

$(b+c)^{2} \geq a^{2}+4(h_{a})^{2}$.

(As usual, a, b, c denote the sides BC, CA, AB respectively.)

Proof.

The given inequality is equivalent to $(b+c)^{2}-a^{2} \geq 4(h_{a})^{2}=\frac{16\triangle^{2}}{a^{2}}$.

where $\triangle$ is the area of triangle ABC. Using the identity

$16\triangle^{2}=[(b+c)^{2}-a^{2}][a^{2}-(b-c)^{2}]$, we see that the inequality to be proved is

$a^{2}-(b-c)^{2} \leq a^{2}$ (here we use $a), which is true. Observe that equality holds iff $b=c$. QED.

More later,

Nalin Pithwa