常用结论

绝对值不等式

| | a | - | b | | \leq | a \pm b | \leq | a | + | b |

\begin{aligned} \sin θ + \sin φ = 2 \sin \frac{θ + φ}{2} \cos \frac{θ - φ}{2} \\ \sin θ - \sin φ = 2 \cos \frac{θ + φ}{2} \sin \frac{θ - φ}{2} \\ \cos θ + \cos φ = 2 \cos \frac{θ + φ}{2} \cos \frac{θ - φ}{2} \\ \cos θ + \cos φ = - 2 \sin \frac{θ + φ}{2} \sin \frac{θ - φ}{2} \\ \sin α \cos β = \frac{\sin (α + β) + \sin (α - β)}{2} \\ \cos α \sin β = \frac{\sin (α + β) - \sin (α - β)}{2} \\ \cos α \cos β = \frac{\cos (α + β) + \cos (α - β)}{2} \\ \sin α \sin β = - \frac{\cos (α + β) - \cos (α + β)}{2} \end{aligned}

\int_{0}^{a} \sqrt{a^{2} - x^{2}} d x = \frac{π}{4} a^{2}

| \int_{a}^{b} f (x) d x | \leq \int_{a}^{b} | f (x) | d x

\int_{0}^{\frac{π}{2}} \sin^{n} x d x = \int_{0}^{\frac{π}{2}} \cos^{n} x d x = {\begin{cases} \frac{(2 m - 1)!!}{(2 m)!!} \frac{π}{2}, n = 2 m \\ \frac{(2 m)!!}{(2 m + 1)!!}, n = 2 m + 1 \end{cases}

k = \frac{| y^{″} |}{(1 + y^{' 2})^{\frac{3}{2}}}

k = \frac{| x^{'} (t) y^{″} (t) - y^{'} (t) x^{″} (t) |}{(x^{'}^{2} (t) + y^{'}^{2} (t))^{\frac{3}{2}}}

k = \frac{| r^{2} + 2 r^{'}^{2} - r r^{″} |}{(r^{2} + r^{'}^{2})^{\frac{3}{2}}}