TITLE: How can I calculate $\sin\left(10^{10^{100}} - 10\right)^\circ$?
QUESTION [15 upvotes]: How can I calculate the sine of a googolplex minus 10 degrees?

REPLY [3 votes]: Consider the sequence $\mu_{n + 1} \equiv 10 \mu_{n} \pmod{360}$ with $\mu_0 = 1$, thus $\mu_n \equiv 10^n \pmod{360}$.
Therefore, $\mu_{n + 3} \equiv 1000 \mu_n \equiv 280 \mu_n \pmod{360}$.
But then, notice that:
$$\mu_{n + 4} \equiv 10 \mu_{n + 3} \equiv 10 \cdot 280 \mu_n \equiv 2800 \mu_n \equiv 280 \mu_n \equiv \mu_{n + 3} \pmod{360}$$
Indeed, this sequence reaches a steady state starting at $n = 3$. Thus, we conclude that:
$$\mu_n \equiv 10^n \equiv 280 \pmod{360} ~ ~ ~ ~ ~ ~ ~ \text{for} ~ n > 2$$

From the above result, it follows that for $n > 2$, the statement below holds:
$$\sin{(10^n - 10)} = \sin{\left ( (10^n - 10) ~ \text{mod} ~ 360 \right )} = \sin{\left ( 280 - 10 \right )} = \sin{(270)} = -1$$