There are six basic **trigonometric functions**: [[Sine]], [[Cosine]], [[Tangent Function]], [[Cosecant]], [[Secant]], [[Cotangent]]. > [!NOTE] Definition ($\pi$) > Contents # Properties > [!info] Derivatives > Using [[Power Series is Termwise Differentiable within Radius of Convergence|derivative of power series]]: $\frac{d}{dx} (\cos x) = -\sin x$ and $\frac{d}{dx} (\sin x ) = \cos x.$ > [!NOTE] Lemma ($\sin x$ is bounded by $x$ on $(0,\pi)$) > If $0\leq x\leq \pi$ then $\sin x \leq x$. ^29bd52 >*Proof*. Let $f(x) = \sin x$. We have $f'(x)=\cos x$ and $f''(x)=-\sin x$. So $f(0)=0,\quad, f'(0)=1$and by [[Taylor's Theorem With Lagrange Remainder for Real Function|Taylor's theorem]] with $n=2$ $f(x)=f(0)+f'(0)x+\frac{f''(t)}{2}x^{2}$This says $\sin x=0+x-\frac{\sin t}{2}x^{2}$for some $t\in(0,x)$. As long as $0\leq x\leq \pi$ we have $0<t<\pi$ and so $\sin t>0$. $\square$ > [!NOTE] Power series of tan > No closed form f > Contents # Applications - [[Chebyshev polynomials]].