Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/27

49. $\scriptstyle {x={\frac {x_{1}+x_{2}}{2}},~y={\frac {y_{1}+y_{2}}{2}}}$; coördinates of middle point.

50. $\scriptstyle {x=x_{0}+x',~y=y_{0}+y'}$; transforming to new origin $\scriptstyle {(x_{0},y_{0})}$.

51. $\scriptstyle {x=x'cos\theta -y'sin\theta ,~y=x'sin\theta +y'cos\theta }$; transforming to new axes making the angle $\scriptstyle {\theta }$ with old.

52. $\scriptstyle {x=\rho \cos \theta ,~y=\rho \sin \theta }$; transforming from rectangular to polar coördinates.

53. $\scriptstyle {\rho ={\sqrt {x^{2}+y^{2}}},~\theta =\operatorname {arc~tan} {\frac {y}{x}}}$; transforming from polar to rectangular coördinates.

54. Different forms of equation of a straight line:

(a) $\scriptstyle {{\frac {y-y_{1}}{x-x_{1}}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$, two-point form;
(b) $\scriptstyle {{\frac {x}{a}}+{\frac {y}{b}}=1}$, intercept form;
(c) $\scriptstyle {y-y_{1}=m(x-x_{1})}$, slope-point form;
(d) $\scriptstyle {y=mx+b}$, slope-intercept form;
(e) $\scriptstyle {x\cos \alpha +y\sin \alpha =p}$, normal form;
(f) $\scriptstyle {Ax+By+C=0}$, general form.

55. $\scriptstyle {\tan \theta ={\frac {m_{1}-m_{2}}{1+m_{1}m_{2}}}}$, angle between two lines whose slopes are $\scriptstyle {m_{1}}$ and $\scriptstyle {m_{2}}$.
⁠$\scriptstyle {m_{1}=m_{2}}$ when lines are parallel,
and⁠$\scriptstyle {m_{1}=-{\frac {1}{m_{2}}}}$ when lines are perpendicular.

56. $\scriptstyle {(x-\alpha )^{2}+(y-\beta )^{2}=r^{2}}$, equation of circle with center $\scriptstyle {(\alpha ,\beta )}$ and radius $\scriptstyle {r}$.

2. Greek alphabet.

Letters	Names	Letters	Names	Letters	Names
$\scriptstyle {\mathrm {A} \quad \alpha }$	Alpha	$\scriptstyle {\mathrm {I} \quad \iota }$	Iota	$\scriptstyle {\mathrm {P} \quad \rho }$	Rho
$\scriptstyle {\mathrm {B} \quad \beta }$	Beta	$\scriptstyle {\mathrm {K} \quad \kappa }$	Kappa	$\scriptstyle {\Sigma \quad \sigma }$	Sigma
$\scriptstyle {\Gamma \quad \gamma }$	Gamma	$\scriptstyle {\Lambda \quad \lambda }$	Lambda	$\scriptstyle {\mathrm {T} \quad \tau }$	Tau
$\scriptstyle {\Delta \quad \delta }$	Delta	$\scriptstyle {\mathrm {M} \quad \mu }$	Mu	$\scriptstyle {\Upsilon \quad \upsilon }$	Upsilon
$\scriptstyle {\mathrm {E} \quad \epsilon }$	Epsilon	$\scriptstyle {\mathrm {N} \quad \nu }$	Nu	$\scriptstyle {\Phi \quad \phi }$	Phi
$\scriptstyle {\mathrm {Z} \quad \zeta }$	Zeta	$\scriptstyle {\Xi \quad \xi }$	Xi	$\scriptstyle {\mathrm {X} \quad \chi }$	Chi
$\scriptstyle {\mathrm {H} \quad \eta }$	Eta	$\scriptstyle {\mathrm {O} \quad \mathrm {o} }$	Omicron	$\scriptstyle {\Psi \quad \psi }$	Psi
$\scriptstyle {\Theta \quad \theta }$	Theta	$\scriptstyle {\Pi \quad \pi }$	Pi	$\scriptstyle {\Omega \quad \omega }$	Omega

3. Rules for signs of the trigonometric functions.

Quadrant	Sin	Cos	Tan	Cot	Sec	Csc
First. . . .	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {+}$
Second. . . .	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$
Third. . . .	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$
Fourth. . . .	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {-}$	$\scriptstyle {-}$	$\scriptstyle {+}$	$\scriptstyle {-}$