Inequalities for $\sin$ and $\cos$

Question

I was working on a problem for math, and in the back of the book, we are given that $|\sin\theta|$ $\leq$ $|\theta|$. It got me thinking, are there any other properties similar to this one that applies to $\cos$ or any of the other trigonometric inequalities?

Answer 1

Observe that

$$\cos\theta\le 1.$$

Then integrating from $0$ to $\theta$,

$$\sin\theta\le\theta.$$

Then integrating from $0$ to $\theta$,

$$1-\cos\theta\le\frac{\theta^2}2.$$

Then integrating from $0$ to $\theta$,

$$\theta-\sin\theta\le\frac{\theta^3}6.$$

Then integrating from $0$ to $\theta$,

$$\frac{\theta^2}2+\cos\theta-1\le\frac{\theta^4}{24}.$$

And so on. This re-establishes the Taylor expansion, with a guaranteed bound. (You can symmetrize using parity arguments.)

Answer 2

Among the well known inequality we can also show that

$$\cos \theta \ge 1-\frac12 \theta^2$$

indeed by trigonometric identities we have that

$$\cos(\theta)=1-2\sin^2 \left(\frac \theta 2\right) \ge 1-2\left(\frac \theta 2\right)^2=1-\frac12 \theta^2$$

and many other similar inequality can be obtained by geometrical consideration or mainly by Taylor's series.

Refer to the related

• How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
• Prove the inequalities $1-\frac{x^2}{2}\le \cos(x)\le1-\frac{x^2}{2}+\frac{x^4}{24}$
• Taylor expansion and trigonometric functions
• Proving that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$