The_Cambridge_Handbook_of_Physics_Formulas

34 Mathematics
Trigonometric and hyperbolic definitions
de Moivre’s theorem (cosx+isinx) n =e inx =cosnx+isinnx (2.215)
cosx = 1 2
(
e ix +e −ix) (2.216) coshx = 1 2
(
e x +e −x) (2.217)
sinx = 1 2i
(
e ix −e −ix) (2.218) sinhx = 1 2
(
e x −e −x) (2.219)
tanx = sinx
cosx
(2.220) tanhx = sinhx
coshx
(2.221)
cosix = coshx (2.222) coshix =cosx (2.223)
sinix = isinhx (2.224) sinhix = isinx (2.225)
cotx =(tanx) −1 (2.226) cothx =(tanhx) −1 (2.227)
secx =(cosx) −1 (2.228) sechx =(coshx) −1 (2.229)
cscx =(sinx) −1 (2.230) cschx = (sinhx) −1 (2.231)
Inverse trigonometric functions a
[ ]
x
arcsinx =arctan
(2.232)
(1−x 2 ) 1/2
[ (1−x 2 ) 1/2 ]
arccosx =arctan
(2.233)
x
[ ]
1
arccscx =arctan
(2.234)
(x 2 −1) 1/2
[
arcsecx =arctan (x 2 −1) 1/2] (2.235)
( 1
arccotx =arctan
x)
(2.236)
1.6
1
0
1.6
1
arccotx
arccosx
arccscx
arcsinx
1
x
arcsecx
arctanx
arccosx = π −arcsinx (2.237)
2
a Valid in the angle range 0 ≤ θ ≤ π/2. Note that arcsinx ≡ sin −1 x etc.
0
1
2 3 4 5
x