Inequalities for $\sin$ and $\cos$ 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?
 A: 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.)
A: 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

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