Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/26

2 Volume of prism$$\scriptstyle{=Ba}$$.

Volume of pyramid$$\scriptstyle{=\frac{1}{3}Ba}$$.

Volume of right circular cylinder$$\scriptstyle{=\pi r^2 a}$$.

Lateral surface of right circular cylinder$$\scriptstyle{=2\pi ra}$$.

Total surface of right circular cylinder$$\scriptstyle{=2\pi r(r + a)}$$.

Volume of right circular cone$$\scriptstyle{=\frac{1}{3}\pi r^3a}$$.

Lateral surface of right circular cone$$\scriptstyle{=\pi rs}$$.

Total surface of right circular cone$$\scriptstyle{=\pi r(r + s)}$$.

Volume of sphere$$\scriptstyle{=\frac{4}{3}\pi r^3}$$.

Surface of sphere$$\scriptstyle{=4\pi r^2}$$.

$$\scriptstyle{\sin x=\frac{1}{\csc x};\ \cos x= \frac{1}{\sec x};\ \tan x=\frac{1}{\cot x}}$$.

$$\scriptstyle{\tan x=\frac{\sin x}{\cos x};\ \cot x=\frac{\cos x}{\sin x}}$$.

$$\scriptstyle{\sin^2 x+\cos^2 x=1;\ 1+\tan^2 x=\sec^2 x;\ 1+\cot^2 x=\csc^2 x}$$. $$\begin{array}{l}\scriptstyle{\sin x=\cos\left(\frac{\pi}{2}-x\right);}\\\scriptstyle{\cos x=\sin\left(\frac{\pi}{2}-x\right);}\\\scriptstyle{\tan x=\cot\left(\frac{\pi}{2}-x\right).}\end{array}$$

$$\begin{array}{l}\scriptstyle{\sin(\pi-x)=\sin x;}\\\scriptstyle{\cos(\pi-x)=-\cos x;}\\\scriptstyle{\tan(\pi-x)=-\tan(x).}\end{array}$$

$$\scriptstyle{\sin(x+y)=\sin x\cos y+\cos x\sin y}$$.

$$\scriptstyle{\sin(x-y)=\sin x\cos y-\cos x\sin y}$$.

$$\scriptstyle{\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y}$$.

$$\scriptstyle{\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}}$$.

$$\scriptstyle{\tan(x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}}$$. $$\scriptstyle{\sin2x=2\sin x\cos x;\ \cos2x=\cos^2x=\sin^2x;\ \tan2x=\frac{2\tan x}{1-\tan^2x}}$$.

$$\scriptstyle{\sin x=2\sin\frac{x}{2}\cos\frac{x}{2};\ \cos x=\cos^2\frac{x}{2}-\sin^2\frac{x}{2};\ \tan x=\frac{2\tan\frac{1}{2}x}{1-\tan^2\frac{1}{2}x}}$$.

$$\scriptstyle{\cos^2x=\frac{1}{2}+\frac{1}{2}\cos2x;\ \sin^2x=\frac{1}{2}-\frac{1}{2}\cos2x}$$.

$$\scriptstyle{1+\cos x=2\cos^2\frac{x}{2};\ 1-\cos x=2\sin^2\frac{x}{2}}$$.

$$\scriptstyle{\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}};\ \cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}};\ \tan\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{1+\cos x}}}$$.

$$\scriptstyle{\sin x+\sin y=2\sin\frac{1}{2}(x+y)\cos\frac{1}{2}(x-y)}$$.

$$\scriptstyle{\sin x-\sin y=2\cos\frac{1}{2}(x+y)\sin\frac{1}{2}(x-y)}$$.

$$\scriptstyle{\cos x+\cos y=2\cos\frac{1}{2}(x+y)\cos\frac{1}{2}(x-y)}$$.

$$\scriptstyle{\cos x-\cos y=-2\sin\frac{1}{2}(x+y)\sin\frac{1}{2}(x-y)}$$.

$$\scriptstyle{\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C};}$$ Law of Sines.

$$\scriptstyle{a^2=b^2+c^2-2bc\cos A;}$$ Law of Cosines.

$$\scriptstyle{d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2};}$$ distance between points $$\scriptstyle{(x_1,y_1)}$$ and $$\scriptstyle{(x_2,y_2)}$$.

$$\scriptstyle{d=\frac{Ax_1+By_1+C}{\pm\sqrt{A^2+B^2}};}$$ distance from line $$\scriptstyle{Ax+By+C=0}$$ to $$\scriptstyle{(x_1,y_1)}$$.