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


  • 2
    $\begingroup$ It is needed to make sure that $\theta$ is positive, so that when you put negative sign next to it, it is surely negative $\endgroup$ – Sonal_sqrt Dec 21 '17 at 5:08
  • $\begingroup$ can not we simply write −θ≤sinθ≤θ &−θ≤1−cosθ≤θ ? $\endgroup$ – user3352074 Dec 29 '17 at 4:22
  • $\begingroup$ In this try putting a negative $\theta$ and it will be clear why we need the absolute sign $\endgroup$ – Sonal_sqrt Dec 29 '17 at 4:25

To make sure that when we are assuming angle to be negative in our inequality , it is negative and we are assuming it to be positive it is positive only as we know that modulus is always positive

  • $\begingroup$ why can not we simply write −θ≤sinθ≤θ and −θ≤1−cosθ≤θ ? $\endgroup$ – user3352074 Dec 29 '17 at 4:22

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!

  • $\begingroup$ Thanks Alex.Let me be clear, So Positive |Θ| means the angel is above the x- axis (measure in counter clockwise) and -|Θ| means the angel is below the x-axis (measure in clockwise), right? $\endgroup$ – user3352074 Dec 21 '17 at 18:30
  • $\begingroup$ why can not we simply write −θ≤sinθ≤θ and −θ≤1−cosθ≤θ ? $\endgroup$ – user3352074 Dec 29 '17 at 4:23
  • $\begingroup$ As I wrote in the answer, try when θ=-0.1. Now it would read 0.1 ≤ sin(-0.1) ≤ -0.1. Clearly false! $\endgroup$ – Alex Meiburg Dec 29 '17 at 5:09
  • $\begingroup$ Thanks Alex.You are right.i understand.−|θ| means the negative of the absolute value of θ. $\endgroup$ – user3352074 Dec 29 '17 at 5:46
  • $\begingroup$ So in Trigonometry,Absolute value describes the distance of an angle on xy-plane from initial ray without considering which direction(clock-wise or anti clock wise) it is.The absolute value of a angle is never negative same as number.Right? $\endgroup$ – user3352074 Dec 29 '17 at 5:55

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