![Part III - Snvhomesnvhome.net/ee-braude/introduction2eo/figures/figures 2... · Spherical waveform coming from a real point source (after [1]). Figure 48. Gaussian-spherical wave](https://static.fdocument.org/doc/165x107/6108a4141d2cbb6d0640185d/part-iii-2-spherical-waveform-coming-from-a-real-point-source-after-1.jpg)
Figures
2
Algebra Geometry Arithmetic a + b c = a c + b c a b + c d = ad + bc bd a b c d = a b d c = ad bc Factoring x 2 − y 2 = (x − y)(x + y) x 3 − y 3 = (x − y)(x 2 + xy + y 2 ) x 3 + y 3 = (x + y)(x 2 − xy + y 2 ) x 4 − y 4 = (x − y)(x + y)(x 2 + y 2 ) Binomial (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 Exponents x n x m = x n+m x n x m = x n−m (x n ) m = x nm x −n = 1 x n (xy) n = x n y n x y n = x n y n x n/m = m √ x n n √ xy = n √ x n √ y n x y = n √ x n √ y Lines Slope m of line through (x 0 , y 0 ) and (x 1 , y 1 ) m = y 1 − y 0 x 1 − x 0 Through (x 0 , y 0 ), slope m y − y 0 = m(x − x 0 ) Slope m, y-intercept b y = mx + b Quadratic Formula If ax 2 + bx + c = 0 then x = −b ± √ b 2 − 4ac 2a Distance Distance d between (x 1 , y 1 ) and (x 2 , y 2 ) d = (x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Triangle Circle Area = 1 2 bh Area = π r 2 c 2 = a 2 + b 2 − 2ab cos θ C = 2π r c a h b r Sector of a Circle Trapezoid Area = 1 2 r 2 θ Area = 1 2 (a + b)h s = r θ (for θ in radians only) r s a b h Sphere Cone Volume = 4 3 π r 3 Volume = 1 3 π r 2 h Surface Area = 4π r 2 Surface Area = π r √ r 2 + h 2 r h r Cylinder Volume = π r 2 h Surface Area = 2π rh r h
-
Upload
drradz-maths -
Category
Documents
-
view
59 -
download
2
Transcript of Figures
P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
GTBL001-front˙end GTBL001-Smith-v16.cls October 17, 2005 20:2
Algebra Geometry
Arithmetica + b
c= a
c+ b
c
a
b+ c
d= ad + bc
bd(a
b
)(
c
d
) =(
a
b
) (d
c
)= ad
bc
Factoringx2 − y2 = (x − y)(x + y) x3 − y3 = (x − y)(x2 + xy + y2)
x3 + y3 = (x + y)(x2 − xy + y2) x4 − y4 = (x − y)(x + y)(x2 + y2)
Binomial(x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2 y + 3xy2 + y3
Exponentsxn xm = xn+m xn
xm= xn−m (xn)m = xnm
x−n = 1
xn(xy)n = xn yn
(x
y
)n
= xn
yn
xn/m = m√
xn n√
xy = n√
x n√
y n
√x
y=
n√
xn√
y
LinesSlope m of line through (x0, y0) and (x1, y1)
m = y1 − y0
x1 − x0
Through (x0, y0), slope m
y − y0 = m(x − x0)
Slope m, y-intercept by = mx + b
Quadratic FormulaIf ax2 + bx + c = 0 then
x = −b ± √b2 − 4ac
2a
DistanceDistance d between (x1, y1) and (x2, y2)
d =√
(x2 − x1)2 + (y2 − y1)2
Triangle CircleArea = 1
2 bh Area = πr2
c2 = a2 + b2 − 2ab cos θ C = 2πr
cah
b
�
r
Sector of a Circle TrapezoidArea = 1
2 r2θ Area = 12 (a + b)h
s = rθ
(for θ in radians only)
�
r
s
a
b
h
Sphere ConeVolume = 4
3 πr3 Volume = 13 πr2h
Surface Area = 4πr2 Surface Area = πr√
r2 + h2
rh
r
CylinderVolume = πr2hSurface Area = 2πrh
r
h
1
P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
GTBL001-front˙end GTBL001-Smith-v16.cls October 17, 2005 20:2
Trigonometry
�
r
(x, y)
�
hyp opp
adj
sin θ = y
r
cos θ = x
r
tan θ = y
x
sin θ = opp
hyp
cos θ = adj
hyp
tan θ = opp
adj
Reciprocals
cot θ = 1
tan θsec θ = 1
cos θcsc θ = 1
sin θ
Definitions
cot θ = cos θ
sin θsec θ = 1
cos θcsc θ = 1
sin θ
Pythagoreansin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ
Cofunctionsin
(π2 − θ
) = cos θ cos(
π2 − θ
) = sin θ tan(
π2 − θ
) = cot θ
Even/Oddsin(−θ ) = −sin θ cos(−θ ) = cos θ tan(−θ ) = −tan θ
Double-Anglesin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ cos 2θ = 1 − 2 sin2 θ
Half-Angle
sin2 θ = 1 − cos 2θ
2cos2 θ = 1 + cos 2θ
2
Additionsin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b − sin a sin b
Subtractionsin(a − b) = sin a cos b − cos a sin b cos(a − b) = cos a cos b + sin a sin b
Sumsin u + sin v = 2 sin
u + v
2cos
u − v
2
cos u + cos v = 2 cosu + v
2cos
u − v
2
Product
sin u sin v = 12 [cos(u − v) − cos(u + v)]
cos u cos v = 12 [cos(u − v) + cos(u + v)]
sin u cos v = 12 [sin(u + v) + sin(u − v)]
cos u sin v = 12 [sin(u + v) − sin(u − v)]
Radians
π
5π/6
3π/4
2π/3π/2
π/3
π/4
π/6
0
sin(0) = 0 cos(0) = 1
sin(
π6
) = 12 cos
(π6
) =√
32
sin(
π4
) =√
22 cos
(π4
) =√
22
sin(
π3
) =√
32 cos
(π3
) = 12
sin(
π2
) = 1 cos(
π2
) = 0
sin( 2π
3
) =√
32 cos
( 2π3
) = − 12
sin( 3π
4
) =√
22 cos
( 3π4
) = −√
22
sin( 5π
6
) = 12 cos
( 5π6
) = −√
32
sin(π ) = 0 cos(π ) = −1
sin(2π ) = 0 cos(2π ) = 1
2
