# Understanding the use of absolute values in the inequalities $-|\theta| \leq \sin\theta \leq |\theta|$ and $-|\theta| \leq 1-\cos\theta \leq |\theta|$

Currently, I am reading Thomas's Calculus. In the Trigonometry Functions section, it is said that the sine and cosine functions satisfy the following inequalities

$$-|\theta| \leq \sin\theta \leq |\theta| \qquad\text{and}\qquad -|\theta| \leq 1-\cos\theta \leq |\theta|$$ for any angle $\theta$ measured in radians

I understand how these are established, but one confusion:

Is the bar "$|\cdot|$" indicating absolute value? Why is the negative sign used before the $|\cdot|$ ?

Best Regards

sabbir

• It is needed to make sure that $\theta$ is positive, so that when you put negative sign next to it, it is surely negative – Sonal_sqrt Dec 21 '17 at 5:08
• can not we simply write −θ≤sinθ≤θ &−θ≤1−cosθ≤θ ? – user3352074 Dec 29 '17 at 4:22
• In this try putting a negative $\theta$ and it will be clear why we need the absolute sign – Sonal_sqrt Dec 29 '17 at 4:25

Yes, the | | is absolute value. Consider that $|x|$ is "$x$ but always positive"; then $-|x|$ is "$x$ but always negative. If you try plugging in, for instance, $\Theta = 0.1$, then sure, the | |'s don't seem that necessary. Try plugging in $\Theta = -0.1$ and you see that, without the | | and -| |, it would be false otherwise!