We are tasked with proving that the absolute value of $\sin(a)$ is less than or equal to the absolute value of $a$, that is:

$|\sin(a)| \leq |a| \quad \text{for all } a \in \mathbb{R}$

1. Case for Small Values of $a$

Let's first consider small values of $a$. From calculus, we know that for values of $a$ close to zero, $\sin(a)$ can be approximated by its Taylor series expansion:

$\sin(a) = a - \frac{a^3}{6} + O(a^5)$

For small $a$, the dominant term is $a$, and higher-order terms are small. This suggests that $|\sin(a)|$ behaves like $|a|$ for small values of $a$, so:

$|\sin(a)| \approx |a|$

As $a \to 0$, we also know that:

$\lim_{a \to 0} \frac{\sin(a)}{a} = 1$

Thus, near $a = 0$, we have $|\sin(a)| \leq |a|$.

2. General Case for All $a$

To prove the inequality for all values of $a$, let's analyze the behavior of $\sin(a)$ and $a$.

• The function $\sin(a)$ is bounded between $-1$ and $1$ for all real numbers $a$:

$-1 \leq \sin(a) \leq 1$

• Therefore, for large values of $a$, $|\sin(a)|$ is always less than or equal to 1. However, $|a|$ grows without bound as $a$ increases. Thus, for large values of $a$, it is clear that $|\sin(a)| \leq |a|$ since $|a| > 1$.

3. Combining the Results

For small values of $a$, we saw that $|\sin(a)| \approx |a|$, and for large values of $a$, $|\sin(a)| \leq 1 \leq |a|$. Therefore, for all values of $a$, the inequality holds:

$|\sin(a)| \leq |a|$

Conclusion

The inequality $|\sin(a)| \leq |a|$ holds for all real numbers $a$, as shown by analyzing the behavior of $\sin(a)$ both near zero and for larger values of $a$.

Would you like any further details or clarification on this proof?

Here are five related questions you might find interesting:

1. What is the Taylor series expansion for $\sin(a)$?
2. How can we rigorously prove that $\lim_{a \to 0} \frac{\sin(a)}{a} = 1$?
3. What other trigonometric functions can be compared to $a$ in terms of growth?
4. How can we use L'Hopital's Rule to confirm limits involving trigonometric functions?
5. Can we use the Mean Value Theorem to provide an alternative proof for this inequality?

Tip: For small angles, $\sin(a)$ can be approximated by $a$, which is the basis for many practical approximations in physics and engineering.