how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$

Solution 1:

If $19^{2013}$ were measured in degrees we could use a simple trick.

$19$ is one more than $18$ which is a multiple of $6$. This means that any power of $19$ is also $1$ more than a multiple of $6$, in particular $19^2 = 361$

$19^2$ being $1$ more than a multiple of $360$ tells us that any even power of $19$ will be 1 more than a multiple of $360$. In particular we can conclude that $19^{2012}$ is one more than a multiple of $360$.

What does that mean? It means that if we travel $19^{2012}$ degrees around the unit circle we will just end up at the $1$ degree tick mark.

$19^{2013}$ is just $19^{2012}$ $19$ times. This means we would make the trip described above $19$ times each time ending up one tick further than the last time meaning that we would finish at $19$ degrees. In other words $19$ degrees is located at the same position of the unit circle as $19^{2013}$.

If you've never worked with modular arithmetic you may not understand why $19^2 = 360 + 1$ implies that $19^{2012} = k \cdot 360+1$.

To see this consider powers of $(360+1)$.

$$360+1$$ $$ (360+1)^2 = 360^2 + 2\cdot360 + 1$$ $$ (360+1)^3 = 360^3 + 3^360^2 + 3*360 + 1$$ $$ \vdots$$ $$ (360+1)^n = 360^n + n *360^{n-1} + \cdots + n * 360 + 1 $$

Notice the result is always one more than a multiple of $360$.

Solution 2:

To find the sine and cosine, you'll need to reduce the angle ($19^{2013} \approx 1.352 \times 10^{2574}$) modulo $2\pi$. So, for reasonable accuracy, you'll need about 2600 digits of $\pi$. Fortunately, the first 100 000 or even million digits are readily accessible online.

It happens that $19^{2013} \approx 1.2329141525482654$ modulo $2\pi$, so:

• $\sin(19^{2013}) \approx \sin(1.2329141525482654) \approx 0.9434588183383549$
• $\cos(19^{2013}) \approx \cos(1.2329141525482654) \approx 0.3314897556480367$

All you need is a programming language that supports arbitrary-precision rational arithmetic, and a way of obtaining lots of digits of $\pi$. A normal TI-89 calculator will provide the former; the hard part is implementing an algorithm for the latter. But that's a topic for another question.